JEE Main & Advanced Mathematics Conic Sections Question Bank

done Self Evaluation Test - Conic Sections Total Questions - 90

• question_answer1)
  If the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] intersects the hyperbola \[xy={{c}^{2}}\] in four points \[P({{x}_{1}},{{y}_{1}}),Q({{x}_{2}},{{y}_{2}}),R({{x}_{3}},{{y}_{3}}),S({{x}_{4}},{{y}_{4}})\] Then
  A)
  \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=0\] done clear
  B)
  \[{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}=2\] done clear
  C)
  \[{{x}_{1}}{{x}_{2}}{{x}_{3}}{{x}_{4}}=2{{c}^{4}}\] done clear
  D)
  \[{{y}_{1}}{{y}_{2}}{{y}_{3}}{{y}_{4}}=2{{c}^{4}}\] done clear
  View Solution play_arrow

• question_answer2)
  Let A be the centre of the circle \[{{x}^{2}}+{{y}^{2}}-2x-4y-20=0,\] and \[B(1,7)\] and \[D(4,-2)\] are points on the circle then, if tangents be drawn at B and D, which meet at C, then area of quadrilateral ABCD is-
  A)
  150 done clear
  B)
  75 done clear
  C)
  75/2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  The equation of a circle which passes through the point (2, 0) and whose centre is the limit of the point of intersection of the lines \[3x+5y=1\] and \[(2+c)x+5{{c}^{2}}y=1\] as c tends to 1, is
  A)
  \[25({{x}^{2}}+{{y}^{2}})+20x+2y-60=0\] done clear
  B)
  \[25({{x}^{2}}+{{y}^{2}})-20x+2y+60=0\] done clear
  C)
  \[25({{x}^{2}}+{{y}^{2}})-20x+2y-60=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer4)
  If the ellipse \[9{{x}^{2}}+16{{y}^{2}}=144\] intercepts the line \[3x+4y=12,\] then what is the length of the chord so formed?
  A)
  5 units done clear
  B)
  6 units done clear
  C)
  8 units done clear
  D)
  10 units done clear
  View Solution play_arrow

• question_answer5)
  If the pair of lines \[a{{x}^{2}}+2(a+b)xy+b{{y}^{2}}=0\] lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then
  A)
  \[3{{a}^{2}}-10ab+3{{b}^{2}}=0\] done clear
  B)
  \[3{{a}^{2}}-2ab+3{{b}^{2}}=0\] done clear
  C)
  \[3{{a}^{2}}+10ab+3{{b}^{2}}=0\] done clear
  D)
  \[3{{a}^{2}}+2ab+3{{b}^{2}}=0\] done clear
  View Solution play_arrow

• question_answer6)
  The line joining (5, 0) to is divided internally in the ratio 2 : 3 at P. If \[\theta \] varies, then the locus of P is
  A)
  A pair of straight lines done clear
  B)
  A circle done clear
  C)
  A straight line done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  If \[x=9\] is the chord of contact of the hyperbola \[{{x}^{2}}-{{y}^{2}}=9\], then the equation of the corresponding pair of tangents is
  A)
  \[9{{x}^{2}}-8{{y}^{2}}+18x-9=0\] done clear
  B)
  \[9{{x}^{2}}-8{{y}^{2}}-18x+9=0\] done clear
  C)
  \[9{{x}^{2}}-8{{y}^{2}}-18x-9=0\] done clear
  D)
  \[9{{x}^{2}}-8{{y}^{2}}+18x+9=0\] done clear
  View Solution play_arrow

• question_answer8)
  If the line \[y=mx+\sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}\] touches the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at the point\[\varphi \]. Then \[\varphi \]=
  A)
  \[{{\sin }^{-1}}(m)\] done clear
  B)
  \[{{\sin }^{-1}}\left( \frac{a}{bm} \right)\] done clear
  C)
  \[{{\sin }^{-1}}\left( \frac{b}{am} \right)\] done clear
  D)
  \[{{\sin }^{-1}}\left( \frac{bm}{a} \right)\] done clear
  View Solution play_arrow

• question_answer9)
  If OA and OB are the tangents form the origin to the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] and C is the centre of the circle, the area of the quadrilateral OACD is
  A)
  \[\frac{1}{2}\sqrt{c({{g}^{2}}+{{f}^{2}}-c)}\] done clear
  B)
  \[\sqrt{c({{g}^{2}}+{{f}^{2}}-c)}\] done clear
  C)
  \[c\sqrt{{{g}^{2}}+{{f}^{2}}-c}\]      done clear
  D)
  \[\frac{\sqrt{{{g}^{2}}+{{f}^{2}}-c}}{c}\] done clear
  View Solution play_arrow

• question_answer10)
  Consider any point P on the ellipse \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{9}=1\] in the first quadrant. Let r and s represent its distances from (4, 0) and (-4, 0) respectively, then (r + s) is equal to
  A)
  10 unit done clear
  B)
  9 unit done clear
  C)
  8 unit done clear
  D)
  6 unit done clear
  View Solution play_arrow

• question_answer11)
  The point \[([P+1],[P])\](where [x] is the greatest integer less than or equal to x), lying inside the region bounded by the circle \[{{x}^{2}}+{{y}^{2}}-2x-15=0\] and \[{{x}^{2}}+{{y}^{2}}-2x-7=0,\]then
  A)
  \[P\in [-1,0)\cup [0,1)\cup [1,2)\] done clear
  B)
  \[P\in [-1,\,\,2)-\{0,\,\,1\}\] done clear
  C)
  \[P\in (-1,\,\,2)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  The line \[y=mx+c\] intersects the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\] at the two real distinct points if
  A)
  \[-r\sqrt{1+{{m}^{2}}}<c<r\sqrt{1+{{m}^{2}}}\] done clear
  B)
  \[-r<c<r\] done clear
  C)
  \[-r\sqrt{1-{{m}^{2}}}<c<r\sqrt{1+{{m}^{2}}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  The equation of the image of circle \[{{x}^{2}}+{{y}^{2}}+16x-24y+183=0\] by the line mirror \[4x+7y+13=0\] is
  A)
  \[{{x}^{2}}+{{y}^{2}}+32x-4y+235=0\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}+32x+4y-235=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}+32x-4y-235=0\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}+32x+4y+235=0\] done clear
  View Solution play_arrow

• question_answer14)
  Tangents at any point on the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] cut the axes at A and B respectively. If the rectangle OAPB (where O is the origin)is completed, then locus of point P is given by :
  A)
  \[\frac{{{a}^{2}}}{{{x}^{2}}}-\frac{{{b}^{2}}}{{{y}^{2}}}=1\] done clear
  B)
  \[\frac{{{a}^{2}}}{{{x}^{2}}}+\frac{{{b}^{2}}}{{{y}^{2}}}=1\] done clear
  C)
  \[\frac{{{a}^{2}}}{{{y}^{2}}}-\frac{{{b}^{2}}}{{{x}^{2}}}=1\] done clear
  D)
  None of these   done clear
  View Solution play_arrow

• question_answer15)
              Let \[P(a\,\,sec\theta ,b\,\,tan\,\theta )\] and Q \[Q(a\,\,sec\,\phi ,\,\,b\,tan\,\,\phi ),\]where \[\theta +\phi =\pi /2,\] be two points on the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1.\] If \[(h,k)\] is the point of intersection of the normal at P and Q, then kz is equal to
  A)
  \[\frac{{{a}^{2}}+{{b}^{2}}}{a}\] done clear
  B)
  \[-\left( \frac{{{a}^{2}}+{{b}^{2}}}{a} \right)\] done clear
  C)
  \[\frac{{{a}^{2}}+{{b}^{2}}}{b}\] done clear
  D)
  \[-\left( \frac{{{a}^{2}}+{{b}^{2}}}{b} \right)\] done clear
  View Solution play_arrow

• question_answer16)
  If the centre of the circle passing through the origin is (3, 4), then the intercepts cut off by the circle on x-axis and y-axis respectively are
  A)
  3 unit and 4 unit done clear
  B)
  6 unit and 4 unit done clear
  C)
  3 unit and 8 unit done clear
  D)
  6 unit and 8 unit done clear
  View Solution play_arrow

• question_answer17)
  Let \[{{S}_{1}},{{S}_{2}}\] be the foci of the ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{8}=1.\]If \[A(x+y)\] is any point on the ellipse, then the maximum area of the triangle \[A{{S}_{1}}{{S}_{2}}\] (in square units) is
  A)
  \[2\sqrt{2}\]       done clear
  B)
  \[2\sqrt{3}\] done clear
  C)
  8 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer18)
  Let S is a circle with centre \[(0,\sqrt{2}).\] Then
  A)
  There cannot be any rational point on S done clear
  B)
  There can be infinitely many rational points on S done clear
  C)
  There can be at most two rational points on S done clear
  D)
  There are exactly two rational points on S done clear
  View Solution play_arrow

• question_answer19)
  Four distinct points \[(2k,3k),\,\,(1,0),\,\,(0,1)\] and \[(0,0)\] lie on a circle for
  A)
  Only one value of k done clear
  B)
  \[0<k<1\] done clear
  C)
  \[k<0\] done clear
  D)
  All integral values of k done clear
  View Solution play_arrow

• question_answer20)
  A hyperbola having the transverse axis of length \[2\sin \theta ,\] is confocal with the ellipse \[3{{x}^{2}}+4{{y}^{2}}=12.\] Then its equation is
  A)
  \[{{x}^{2}}\cos e{{c}^{2}}\theta -{{y}^{2}}{{\sec }^{2}}\theta =1\] done clear
  B)
  \[{{x}^{2}}{{\sec }^{2}}\theta -{{y}^{2}}\cos e{{c}^{2}}\theta =1\] done clear
  C)
  \[{{x}^{2}}{{\sin }^{2}}\theta -{{y}^{2}}{{\cos }^{2}}\theta =1\] done clear
  D)
  \[{{x}^{2}}{{\cos }^{2}}\theta -{{y}^{2}}{{\sin }^{2}}\theta =1\] done clear
  View Solution play_arrow

• question_answer21)
  The value of m, for which the line \[y=mx+\frac{25\sqrt{3}}{3}\] is a normal to the conic \[\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{9}=1,\] is
  A)
  \[-\frac{2}{\sqrt{3}}\] done clear
  B)
  \[\sqrt{3}\] done clear
  C)
  \[-\frac{\sqrt{3}}{2}\]     done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer22)
  In the given figure, the equation of the larger circle is \[{{x}^{2}}+{{y}^{2}}+4y-5=0\] and the distance between centres is 4. Then the equation of smaller circle is
  A)
  \[{{(x-\sqrt{7})}^{2}}+{{(y-1)}^{2}}=1\] done clear
  B)
  \[{{(x+\sqrt{7})}^{2}}+{{(y-1)}^{2}}=1\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}=2\sqrt{7}x+2y\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer23)
  The angle of intersection of the circles \[{{x}^{2}}+{{y}^{2}}=4\] and \[{{x}^{2}}+{{y}^{2}}=2x+2y\] is
  A)
  \[\frac{\pi }{2}\] done clear
  B)
  \[\frac{\pi }{3}\] done clear
  C)
  \[\frac{\pi }{6}\] done clear
  D)
  \[\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer24)
  The hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] passes through the point \[(3\sqrt{5},1)\] and the length of its laths rectum is \[\frac{4}{3}\]units. The length of the conjugate axis is
  A)
  2 units done clear
  B)
  3 units done clear
  C)
  4 units done clear
  D)
  5 units done clear
  View Solution play_arrow

• question_answer25)
  A line is drawn through a fixed point \[P(\alpha ,\beta )\] to cut the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] at A and B, then PA.PB is equal to
  A)
  \[{{\alpha }^{2}}+{{\beta }^{2}}\] done clear
  B)
  \[{{\alpha }^{2}}+{{\beta }^{2}}-{{\alpha }^{2}}\] done clear
  C)
  \[{{\alpha }^{2}}\] done clear
  D)
  \[{{\alpha }^{2}}+{{\beta }^{2}}+{{\alpha }^{2}}\] done clear
  View Solution play_arrow

• question_answer26)
  If the line \[x+y=1\] is a tangent to a circle with centre (2, 3), then its equation is
  A)
  \[{{x}^{2}}+{{y}^{2}}+2x+2y+5=0\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}-4x-6y+5=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}-x-y+3=0\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}+5x+2y=0\] done clear
  View Solution play_arrow

• question_answer27)
  Distances form the origin to the centres of the three circles \[{{x}^{2}}+{{y}^{2}}-2{{\lambda }_{i}}x={{c}^{2}}\] (where c is constant and i= 1, 2, 3) are in GP. Then the lengths of tangents drawn from any point on the circle \[{{x}^{2}}+{{y}^{2}}={{c}^{2}}\] to these circles are in
  A)
  A.P. done clear
  B)
  GP. done clear
  C)
  H.P. done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer28)
  If AB is a double ordinate of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] such that \[\Delta OAB\] is an equilateral triangle O being the origin, then the eccentricity of the hyperbola satisfies.
  A)
  \[e>\sqrt{3}\] done clear
  B)
  \[1<e<\frac{2}{\sqrt{3}}\] done clear
  C)
  \[e=\frac{2}{\sqrt{3}}\] done clear
  D)
  \[e>\frac{2}{\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer29)
  The length of the chord intercepted by the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\] on the line \[\frac{x}{a}+\frac{y}{b}=1\] is
  A)
  \[\sqrt{\frac{{{r}^{2}}({{a}^{2}}+{{b}^{2}})-{{a}^{2}}{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}}\] done clear
  B)
  \[2\sqrt{\frac{{{r}^{2}}({{a}^{2}}+{{b}^{2}})-{{a}^{2}}{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}}\] done clear
  C)
  \[2\frac{\sqrt{{{r}^{2}}({{a}^{2}}+{{b}^{2}})-{{a}^{2}}{{b}^{2}}}}{{{a}^{2}}+{{b}^{2}}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer30)
  Area of the equilateral triangle inscribed in the circle \[{{x}^{2}}+{{y}^{2}}-7x+9y+5=0\] is
  A)
  \[\frac{155}{8}\sqrt{3}\] square units done clear
  B)
  \[\frac{165}{8}\sqrt{3}\] square units done clear
  C)
  \[\frac{175}{8}\sqrt{3}\] square units done clear
  D)
  \[\frac{165}{8}\sqrt{3}\] square units done clear
  View Solution play_arrow

• question_answer31)
  If \[{{\left( \frac{x}{a} \right)}^{2}}+\left( {{\frac{y}{b}}^{2}} \right)=1(a>b)\] and \[{{x}^{2}}-{{y}^{2}}={{c}^{2}}\] cut at right angles, then
  A)
  \[{{a}^{2}}+{{b}^{2}}=2{{c}^{2}}\]         done clear
  B)
  \[{{b}^{2}}-{{a}^{2}}=2{{c}^{2}}\] done clear
  C)
  \[{{a}^{2}}-{{b}^{2}}=2{{c}^{2}}\] done clear
  D)
  \[{{a}^{2}}-{{b}^{2}}={{c}^{2}}\] done clear
  View Solution play_arrow

• question_answer32)
  The common chord of \[{{x}^{2}}+{{y}^{2}}-4x-4y=0\] and \[{{x}^{2}}+{{y}^{2}}=16\] subtends at the origin an angle equal to
  A)
  \[\frac{\pi }{6}\] done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{3}\] done clear
  D)
  \[\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer33)
  If p is the length of the perpendicular form the focus S of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] to a tangent at a point P on the ellipse, then \[\frac{2a}{SP}-1=\]
  A)
  \[\frac{{{a}^{2}}}{{{p}^{2}}}\] done clear
  B)
  \[\frac{{{b}^{2}}}{{{p}^{2}}}\] done clear
  C)
  \[{{p}^{2}}\] done clear
  D)
  \[\frac{{{a}^{2}}+{{b}^{2}}}{{{p}^{2}}}\] done clear
  View Solution play_arrow

• question_answer34)
  If the line \[x\cos \alpha +y\sin \alpha =p\] represents the common chord of the circles \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] and \[{{x}^{2}}+{{y}^{2}}+{{b}^{2}}(a>b),\] where A and B lie on the first circle and P and Q lie on the second circle, then AP is equal to
  A)
  \[\sqrt{{{a}^{2}}+{{p}^{2}}}+\sqrt{{{b}^{2}}+{{p}^{2}}}\] done clear
  B)
  \[\sqrt{{{a}^{2}}-{{p}^{2}}}+\sqrt{{{b}^{2}}-{{p}^{2}}}\] done clear
  C)
  \[\sqrt{{{a}^{2}}-{{p}^{2}}}-\sqrt{{{b}^{2}}-{{p}^{2}}}\] done clear
  D)
  \[\sqrt{{{a}^{2}}+{{p}^{2}}}-\sqrt{{{b}^{2}}+{{p}^{2}}}\] done clear
  View Solution play_arrow

• question_answer35)
  If the circles \[{{x}^{2}}+{{y}^{2}}+2ax+cy+a=0\] and \[{{x}^{2}}+{{y}^{2}}-3ax+dy-1=0\] intersect in two distinct points P and Q then the line \[5x+by-a=0\]passes through P and Q for
  A)
  Exactly one value of a done clear
  B)
  No value of a done clear
  C)
  Infinitely many values of a done clear
  D)
  Exactly two values of a done clear
  View Solution play_arrow

• question_answer36)
  The length of the chord \[x+y=3\] intercepted by the circle \[{{x}^{2}}+{{y}^{2}}-2x-2y-2=0\] is
  A)
  \[\frac{7}{2}\] done clear
  B)
  \[\frac{3\sqrt{3}}{2}\] done clear
  C)
  \[\sqrt{14}\] done clear
  D)
  \[\frac{\sqrt{7}}{2}\] done clear
  View Solution play_arrow

• question_answer37)
  A tangent to the parabola \[{{y}^{2}}=8x,\] which makes an angle of \[45{}^\circ \] with the straight line \[y=3x+5\]is
  A)
  \[2x-y+1=0\] done clear
  B)
  \[2x+y+1=0\] done clear
  C)
  \[x-2y+8=0\] done clear
  D)
  Both & done clear
  View Solution play_arrow

• question_answer38)
  Equation of the latus rectum of the hyperbola\[{{(10x-5)}^{2}}+{{(10y-2)}^{2}}=9{{(3x+4y-7)}^{2}}\] is
  A)
  \[y-\frac{1}{5}=-\frac{3}{4}\left( x-\frac{1}{2} \right)\] done clear
  B)
  \[x-\frac{1}{5}=-\frac{3}{4}\left( y-\frac{1}{2} \right)\] done clear
  C)
  \[y+\frac{1}{5}=-\frac{3}{4}\left( x+\frac{1}{2} \right)\] done clear
  D)
  \[x+\frac{1}{5}=-\frac{3}{4}\left( y+\frac{1}{2} \right)\] done clear
  View Solution play_arrow

• question_answer39)
  The line passing through the extremity A of the major axis and the extremity B of the minor axis of the ellipse \[{{x}^{2}}+9{{y}^{2}}=9\] meets its auxiliary circle at the point M. Then the area of the triangle with vertices A, M. and the origin O is
  A)
  \[\frac{31}{10}\] done clear
  B)
  \[\frac{29}{10}\] done clear
  C)
  \[\frac{21}{10}\] done clear
  D)
  \[\frac{27}{10}\] done clear
  View Solution play_arrow

• question_answer40)
  If polar of a circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] with respect to \[(x',y')\] is \[Ax+By+C=0,\] then its pole will be
  A)
  \[\left( \frac{{{a}^{2}}A}{-C},\frac{{{a}^{2}}B}{-C} \right)\] done clear
  B)
  \[\left( \frac{{{a}^{2}}A}{C},\frac{{{a}^{2}}B}{C} \right)\] done clear
  C)
  \[\left( \frac{{{a}^{2}}C}{A},\frac{{{a}^{2}}C}{B} \right)\] done clear
  D)
  \[\left( \frac{{{a}^{2}}C}{-A},\frac{{{a}^{2}}C}{-B} \right)\] done clear
  View Solution play_arrow

• question_answer41)
  If tangents are drawn from any point on the line \[x+4a=0\] to the parabola \[{{y}^{2}}=4ax,\] then their chord of contact subtends angle at the vertex equal to
  A)
  \[\frac{\pi }{4}\] done clear
  B)
  \[\frac{\pi }{3}\] done clear
  C)
  \[\frac{\pi }{2}\] done clear
  D)
  \[\frac{\pi }{6}\] done clear
  View Solution play_arrow

• question_answer42)
  The locus of the point of intersection of two tangents to the parabola \[{{y}^{2}}=4ax,\] which are at right angle to one another is
  A)
  \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] done clear
  B)
  \[a{{y}^{2}}=x\] done clear
  C)
  \[x+a=0\] done clear
  D)
  \[x+y\pm a=0\] done clear
  View Solution play_arrow

• question_answer43)
  If from any point P, tangents PT, PT? are drawn to two given circles with centres A and B respectively; and if PN is the perpendicular form P on their radical axis, then \[P{{T}^{2}}-PT{{'}^{2}}=\]
  A)
  PN.AB done clear
  B)
  \[2PN.AB\] done clear
  C)
  \[4PN.AB\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer44)
  The number of points (a, b) where a and b are positive integers lying on the hyperbola \[{{x}^{2}}-{{y}^{2}}=512\] is
  A)
  3 done clear
  B)
  4 done clear
  C)
  5 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer45)
  Let d be the perpendicular distance from the centre of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] to the tangent drawn at a point P on the ellipse. If \[{{F}_{1}}\] and \[{{F}_{2}}\] be the foci of the ellipse, then \[{{(P{{F}_{1}}-P{{F}_{2}})}^{2}}=\]
  A)
  \[4{{a}^{2}}\left( 1-\frac{{{b}^{2}}}{{{d}^{2}}} \right)\] done clear
  B)
  \[{{a}^{2}}\left( 1-\frac{{{b}^{2}}}{{{d}^{2}}} \right)\] done clear
  C)
  \[4{{a}^{2}}\left( 1-\frac{{{a}^{2}}}{{{d}^{2}}} \right)\] done clear
  D)
  \[{{b}^{2}}\left( 1-\frac{{{a}^{2}}}{{{d}^{2}}} \right)\] done clear
  View Solution play_arrow

• question_answer46)
  Through the vertex O at a parabola \[{{y}^{2}}=4x,\] chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ is
  A)
  \[{{y}^{2}}=2x+8\] done clear
  B)
  \[{{y}^{2}}=x+8\] done clear
  C)
  \[{{y}^{2}}=2x-8\] done clear
  D)
  \[{{y}^{2}}=x-8\] done clear
  View Solution play_arrow

• question_answer47)
  If the equation of the common tangent at the point \[(1,-1)\] to the two circles, each of radius 13, is \[12x+5y-7=0\] then the centres of the two circles are
  A)
  \[(13,4),(-11,6)\] done clear
  B)
  \[(13,4),(-11,-6)\] done clear
  C)
  \[(13,-4),(-11,-6)\] done clear
  D)
  \[(-13,4),(-11,-6)\] done clear
  View Solution play_arrow

• question_answer48)
  The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is:
  A)
  \[\frac{4}{3}\] done clear
  B)
  \[\frac{4}{\sqrt{3}}\] done clear
  C)
  \[\frac{2}{\sqrt{3}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer49)
  If tangents are drawn to the parabola \[{{y}^{2}}=4ax\]at points whose abscissae are in the ratio \[{{m}^{2}}:1,\] then the locus of their point of intersection is the curve \[\left( m>0 \right)\]
  A)
  \[{{y}^{2}}={{({{m}^{1/2}}-{{m}^{-1/2}})}^{2}}ax\] done clear
  B)
  \[{{y}^{2}}={{({{m}^{1/2}}+{{m}^{-1/2}})}^{2}}ax\] done clear
  C)
  \[{{y}^{2}}={{({{m}^{1/2}}+{{m}^{-1/2}})}^{2}}x\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer50)
  What is the area of the triangle formed by the lines joining the vertex of the parabola \[{{x}^{2}}=12y\] to the ends of the latus rectum?
  A)
  9 square units done clear
  B)
  12 square units done clear
  C)
  14 square units done clear
  D)
  18 square units done clear
  View Solution play_arrow

• question_answer51)
  If a variable point P on an ellipse of eccentricity e lines joining the foci \[{{S}_{1}}\] and \[{{S}_{2}}\] then the in centre of the triangle \[P{{S}_{1}}{{S}_{2}}\] lies on
  A)
  The major axis of the ellipse done clear
  B)
  The circle with radius e done clear
  C)
  Another ellipse of eccentricity \[\sqrt{\frac{3+{{e}^{2}}}{4}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer52)
  If the eccentricity of the hyperbola \[{{x}^{2}}-{{y}^{2}}{{\sec }^{2}}\theta =4\] is \[\sqrt{3}\] times the eccentricity of the ellipse \[{{x}^{2}}{{\sec }^{2}}\theta +{{y}^{2}}=16.\] then the value of \[\theta \] equals.
  A)
  \[\frac{\pi }{6}\] done clear
  B)
  \[\frac{3\pi }{4}\]           done clear
  C)
  \[\frac{\pi }{3}\] done clear
  D)
  \[\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer53)
  Consider a circle of radius R. what is the length of a chord which subtends an angle \[\theta \] at the centre?
  A)
  \[2R\sin \left( \frac{\theta }{2} \right)\] done clear
  B)
  \[2R\sin \theta \] done clear
  C)
  \[2R\tan \left( \frac{\theta }{2} \right)\] done clear
  D)
  \[2R\tan \theta \] done clear
  View Solution play_arrow

• question_answer54)
  The limiting points of the coaxial system determined by the circles \[{{x}^{2}}+{{y}^{2}}-2x-6y+9=0\] and \[{{x}^{2}}+{{y}^{2}}+6x-2y+1=0\]
  A)
  \[(-1,2),\left( \frac{3}{5},\frac{-14}{5} \right)\] done clear
  B)
  \[(-1,2),\left( \frac{3}{5},\frac{14}{5} \right)\] done clear
  C)
  \[(-1,2),\left( \frac{-3}{5},\frac{14}{5} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer55)
  If the chords of contact of tangents from two points \[(\alpha ,\beta )\] and \[(\gamma ,\delta )\] to the ellipse \[\frac{{{x}^{2}}}{5}+\frac{{{y}^{2}}}{2}=1\] are perpendicular, then \[\frac{\alpha \gamma }{\beta \delta }=\]
  A)
  \[\frac{4}{25}\] done clear
  B)
  \[\frac{-4}{25}\] done clear
  C)
  \[\frac{25}{4}\] done clear
  D)
  \[\frac{-25}{4}\] done clear
  View Solution play_arrow

• question_answer56)
  A line PQ meets the parabola \[{{y}^{2}}-4ax\] in R such that PQ is bisected at R. if the coordinates of P are \[({{x}_{1}},{{y}_{1}})\] then the locus of Q is the parabola
  A)
  \[{{(y+{{y}_{1}})}^{2}}=8a(x+{{x}_{1}})\] done clear
  B)
  \[{{(y-{{y}_{1}})}^{2}}=8a(x+{{x}_{1}})\] done clear
  C)
  \[{{(y+{{y}_{1}})}^{2}}=8a(x-{{x}_{1}})\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer57)
  The equation of the ellipse with its centre at \[(1,2),\] focus at (6,2) and passing through the point \[(4,6)\]is \[\frac{{{(x-1)}^{2}}}{{{a}^{2}}}+\frac{{{(y-2)}^{2}}}{{{b}^{2}}}=1\], then
  A)
  \[{{a}^{2}}=1,{{b}^{2}}=25\] done clear
  B)
  \[{{a}^{2}}=25,{{b}^{2}}=20\] done clear
  C)
  \[{{a}^{2}}=20,{{b}^{2}}=25\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer58)
  A point moves such that the square of its distance from a straight line is equal to the difference between the square of it distance from the centre of a circle and the square of the radius of the circle. The locus of the point is
  A)
  A straight line at right angle to the given line done clear
  B)
  A circle concentric with the given circle done clear
  C)
  A parabola with its axis parallel to the given line done clear
  D)
  A parabola with its axis perpendicular to the given line done clear
  View Solution play_arrow

• question_answer59)
  The point \[(a,2a)\] is an interior point of region bounded by the parabola \[{{x}^{2}}=16y\] and the double ordinate through focus then
  A)
  a<4      done clear
  B)
  0<a<4 done clear
  C)
  0<a<2 done clear
  D)
  a>4 done clear
  View Solution play_arrow

• question_answer60)
  If the tangents at P and Q on a parabola meet in T, Then SP, ST and SQ are in
  A)
  A.P done clear
  B)
  GP. done clear
  C)
  H.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer61)
  An ellipse has OB as semi minor axis, F and F? its foci and the angle FBF? is a right angle. Then the eccentricity of the ellipse is
  A)
  \[\frac{1}{\sqrt{2}}\] done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[\frac{1}{4}\] done clear
  D)
  \[\frac{1}{\sqrt{3}}\]   done clear
  View Solution play_arrow

• question_answer62)
  The normal at the point \[(b{{t}^{2}}_{1},2b{{t}_{1}})\] on a parabola meets the parabola again in the point \[(b{{t}^{2}}_{2},2b{{t}_{2}})\] Then
  A)
  \[{{t}_{2}}={{t}_{1}}+\frac{2}{{{t}_{1}}}\] done clear
  B)
  \[{{t}_{2}}=-{{t}_{1}}-\frac{2}{{{t}_{1}}}\] done clear
  C)
  \[{{t}_{2}}=-{{t}_{1}}+\frac{2}{{{t}_{1}}}\] done clear
  D)
  \[{{t}_{2}}={{t}_{1}}-\frac{2}{{{t}_{1}}}\] done clear
  View Solution play_arrow

• question_answer63)
  Let \[{{S}_{1}},{{S}_{2}}\] be the foci of the ellipse, \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{8}=1\]. If \[A(x+y)\] is any point on the ellipse, then the maximum area of the triangle \[A{{S}_{1}}{{S}_{2}}\] (in square units) is
  A)
  \[2\sqrt{2}\] done clear
  B)
  \[2\sqrt{3}\] done clear
  C)
  8 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer64)
  The equation of the parabola whose focus is (0, 0) and the tangent at the vertex is \[8x-y+1=0\] is
  A)
  \[{{x}^{2}}+{{y}^{2}}+2xy-4x+4y-4=0\] done clear
  B)
  \[{{x}^{2}}-4x+4y-4=0\] done clear
  C)
  \[{{y}^{2}}-4x+4y-4=0\] done clear
  D)
  \[2{{x}^{2}}+2{{y}^{2}}-4xy-x+y-4=0\] done clear
  View Solution play_arrow

• question_answer65)
  If \[P(\theta )\] and \[Q\left( \frac{\pi }{2}+\theta  \right)\] are two points on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then locus of the mid-point of PQ is
  A)
  \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=\frac{1}{2},\] done clear
  B)
  \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=4\] done clear
  C)
  \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer66)
  If \[P\equiv (x,y),{{F}_{1}}\equiv (3,0),{{F}_{2}}\equiv (-3,0)\] and \[16{{x}^{2}}+25{{y}^{2}}=400,\] then \[P{{F}_{1}}+P{{F}_{2}}\] equals
  A)
  8 done clear
  B)
  6 done clear
  C)
  10 done clear
  D)
  12 done clear
  View Solution play_arrow

• question_answer67)
  Equation of the parabola whose vertex is \[(-3,-2),\] axis is horizontal and which passes through the point \[(1,2)\] is
  A)
  \[{{y}^{2}}+4y+4x-8=0\] done clear
  B)
  \[{{y}^{2}}+4y-4x+8=0\] done clear
  C)
  \[{{y}^{2}}+4y-4x-8=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer68)
  If a Point \[P(x,y)\] moves along the ellipse \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\] and if C is the centre of the ellipse, then, 4 max \[\{CP\}+5min\{CP\}=\]
  A)
  25 done clear
  B)
  40 done clear
  C)
  45 done clear
  D)
  54 done clear
  View Solution play_arrow

• question_answer69)
  The equation of one of the common tangents to the parabola \[{{y}^{2}}=8x\] and \[{{x}^{2}}+{{y}^{2}}-12x+4=0\]is
  A)
  \[y=-x+2\] done clear
  B)
  \[y=x-2\] done clear
  C)
  \[y=x+2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer70)
  An equilateral triangle is inscribed in the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]with one of the vertices at (a, 0). What is the equation of the side opposite to this vertex?
  A)
  \[2x-a=0\] done clear
  B)
  \[x+a=0\] done clear
  C)
  \[2x+a=0\] done clear
  D)
  \[3x-2a=0\] done clear
  View Solution play_arrow

• question_answer71)
  What is the equation to circle which touches both the axes and has centre on the line \[x+y=4?\]
  A)
  \[{{x}^{2}}+{{y}^{2}}-4x+4y+4=0\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}-4x-4y+4=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}+4x-4y-4=0\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}+4x+4y-4=0\] done clear
  View Solution play_arrow

• question_answer72)
  If the coordinates of four concyclie points on the rectangular hyperbola \[xy={{c}^{2}}\] are \[(c{{t}_{i}},\,\,c/{{t}_{i}}),i=1,\,\,2,\,\,3,\,\,4\] then
  A)
  \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=-1\] done clear
  B)
  \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=1\] done clear
  C)
  \[{{t}_{1}}{{t}_{3}}={{t}_{2}}{{t}_{4}}\] done clear
  D)
  \[{{t}_{1}}+{{t}_{2}}+{{t}_{3}}+{{t}_{4}}={{c}^{2}}\] done clear
  View Solution play_arrow

• question_answer73)
  If the angle between the straight lines joining the foci to an extremity of minor axis in an ellipse be \[90{}^\circ \]; then the eccentricity of the ellipse is
  A)
  \[\frac{1}{2}\] done clear
  B)
  \[\frac{1}{\sqrt{3}}\] done clear
  C)
  \[\frac{1}{\sqrt{2}}\] done clear
  D)
  \[\frac{1}{3}\] done clear
  View Solution play_arrow

• question_answer74)
  If two circles A, B of equal radii pass through the centres of each other, then what is the ratio of the length of the smaller are to the circumference of the circle A cut off by the circle B?
  A)
  \[\frac{1}{2}\] done clear
  B)
  \[\frac{1}{4}\] done clear
  C)
  \[\frac{1}{3}\] done clear
  D)
  \[\frac{2}{3}\] done clear
  View Solution play_arrow

• question_answer75)
  The equation of the circle which touches the axes at a distance 5 from the origin is \[{{y}^{2}}+{{x}^{2}}-2ax-2ay+{{a}^{2}}=0.\] what is the value of\[\alpha \]?
  A)
  4 done clear
  B)
  5 done clear
  C)
  6 done clear
  D)
  7 done clear
  View Solution play_arrow

• question_answer76)
  The curve represented by\[x=2(cost+sint),,y=5(cost-sint)\] is
  A)
  A circle done clear
  B)
  A parabola done clear
  C)
  An ellipse done clear
  D)
  A hyperbola done clear
  View Solution play_arrow

• question_answer77)
  Under which one of the following conditions does the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]meet the x-axis in two points on opposite sides of the origin?
  A)
  \[c>0\] done clear
  B)
  \[c<0\] done clear
  C)
  \[c=0\]  done clear
  D)
  \[c\le 0\] done clear
  View Solution play_arrow

• question_answer78)
  If the focal distance of an end of the minor axis of any ellipse (referred to its axis as the axes of x and y respectively) is k and the distance between the foci is 2h, then its equation is
  A)
  \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{k}^{2}}+{{h}^{2}}}=1\] done clear
  B)
  \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{h}^{2}}-{{k}^{2}}}=1\] done clear
  C)
  \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{k}^{2}}-{{h}^{2}}}=1\] done clear
  D)
  \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{h}^{2}}}=1\] done clear
  View Solution play_arrow

• question_answer79)
  If the latus rectum of an ellipse is equal to one half its minor axis, what is the eccentricity of the ellipse?
  A)
  \[\frac{1}{2}\] done clear
  B)
  \[\frac{\sqrt{3}}{2}\] done clear
  C)
  \[\frac{3}{4}\] done clear
  D)
  \[\frac{\sqrt{15}}{4}\] done clear
  View Solution play_arrow

• question_answer80)
  The sum of the focal distances of  a point on the ellipse \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{9}=1\] is:
  A)
  4 units done clear
  B)
  6 units done clear
  C)
  8 units done clear
  D)
  10 units done clear
  View Solution play_arrow

• question_answer81)
  Equation of the hyperbola whose directirx is \[2x+y=1\], focus (1, 2) and eccentricity \[\sqrt{3}\] is
  A)
  \[7{{x}^{2}}-2{{y}^{2}}+12xy-2x+14y-22=0\] done clear
  B)
  \[5{{x}^{2}}-2{{y}^{2}}+10xy+2x+5y-20=0\] done clear
  C)
  \[4{{x}^{2}}+8{{y}^{2}}+8xy+2x-2y+10=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer82)
  The sum of the squares of the perpendiculars on any tangent to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] from two points on the minor axis each at a distance \[\sqrt{{{a}^{2}}-{{b}^{2}}}\] from the centre is
  A)
  \[2{{a}^{2}}\] done clear
  B)
  \[2{{b}^{2}}\] done clear
  C)
  \[{{a}^{2}}+{{b}^{2}}\] done clear
  D)
  \[{{a}^{2}}-{{b}^{2}}\] done clear
  View Solution play_arrow

• question_answer83)
  A point on the ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{9}=1\] at a distance equal to the mean of the lengths of the semi-major axis and semi-minor axis from the centre is
  A)
  \[\left( \frac{2\sqrt{91}}{7},\frac{3\sqrt{105}}{14} \right)\] done clear
  B)
  \[\left( \frac{2\sqrt{91}}{7}-\frac{3\sqrt{105}}{14} \right)\] done clear
  C)
  \[\left( \frac{2\sqrt{105}}{7},\frac{3\sqrt{91}}{14} \right)\] done clear
  D)
  \[\left( -\frac{2\sqrt{105}}{7}-\frac{3\sqrt{91}}{14} \right)\] done clear
  View Solution play_arrow

• question_answer84)
  Consider the parabolas \[{{S}_{1}}\equiv {{y}^{2}}-4ax=0\] and \[{{S}_{2}}\equiv {{y}^{2}}-4bx=0.\] \[{{S}_{2}}\] will contain \[{{S}_{1}},\] if
  A)
  \[a>b>0\] done clear
  B)
  \[b>a>0\] done clear
  C)
  \[a>0,\,\,b<0\,\,but\,\,\left| \,b\, \right|>a\] done clear
  D)
  \[a<0,\,\,b>0\,\,but\,\,b>\left| \,a\, \right|\] done clear
  View Solution play_arrow

• question_answer85)
  What are the points of intersection of the curve\[4{{x}^{2}}-9{{y}^{2}}=1\] with its conjugate axis?
  A)
  \[(1/2,0)\] and \[(-1/2,0)\] done clear
  B)
  \[(0,2)\] and \[(0,-2)\] done clear
  C)
  \[(0,3)\] and \[(0,-3)\] done clear
  D)
  No such point exists done clear
  View Solution play_arrow

• question_answer86)
  The curve described parametrically by \[x=2-3\sec t,y=1+4\tan t\] represents:
  A)
  An ellipse centred at (2, 1) and of eccentricity \[\frac{3}{5}\] done clear
  B)
  A circle centred at (2, 1) and of radius 5 units done clear
  C)
  A hyperbola centred at (2, 1) & of eccentricity \[\frac{8}{5}\] done clear
  D)
  A hyperbola centred at \[(2,1)\]& of eccentricity \[\frac{5}{3}\] done clear
  View Solution play_arrow

• question_answer87)
  The locus of the point of intersection of two tangents of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] which are inclined at angles \[{{\theta }_{1}}\], and \[{{\theta }_{2}}\] with the major axis such that \[{{\tan }^{2}}{{\theta }_{1}}+{{\tan }^{2}}{{\theta }_{2}}\] is constant, is
  A)
  \[4{{x}^{2}}{{y}^{2}}+2({{x}^{2}}-{{a}^{2}})({{y}^{2}}-{{b}^{2}})=k{{({{x}^{2}}-{{a}^{2}})}^{2}}\] done clear
  B)
  \[4{{x}^{2}}{{y}^{2}}-2({{x}^{2}}-{{a}^{2}})({{y}^{2}}-{{b}^{2}})=k{{({{x}^{2}}-{{a}^{2}})}^{2}}\] done clear
  C)
  \[4{{x}^{2}}{{y}^{2}}-2({{x}^{2}}-{{a}^{2}})({{y}^{2}}-{{b}^{2}})=k{{({{x}^{2}}+{{a}^{2}})}^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer88)
  Let E be the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1\] and C be the circle \[{{x}^{2}}+{{y}^{2}}=9.\] Let \[P=(1,2)\] and \[Q=(2,1)\] Which one of the following is correct?
  A)
  Q lies inside C but outside E done clear
  B)
  Q lies outside both C and E done clear
  C)
  P lies inside both C and E done clear
  D)
  P lies inside both C but outside E. done clear
  View Solution play_arrow

• question_answer89)
  Which one of the following is correct? The eccentricity of the conic \[\frac{{{x}^{2}}}{{{a}^{2}}+\lambda }+\frac{{{y}^{2}}}{{{b}^{2}}+\lambda }=1,(\lambda \ge 0)\]
  A)
  Increases with increase in \[\lambda \] done clear
  B)
  Decreases with increase in \[\lambda \] done clear
  C)
  Does not change with \[\lambda \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer90)
  What is the area of the triangle formed by the lines joining the vertex of the parabola \[{{x}^{2}}=12y\] to the ends of the latus rectum?
  A)
  9 square units done clear
  B)
  12 square units done clear
  C)
  14 square units done clear
  D)
  18 square units done clear
  View Solution play_arrow