JEE Main & Advanced Mathematics Straight Line Question Bank

done Self Evaluation Test - Straight Lines & Pair Striaght Lines Total Questions - 70

• question_answer1)
  Area of the triangle formed by the line \[x+y=3\]and the angle bisectors of the pairs of straight lines \[{{x}^{2}}-{{y}^{2}}+2y=1\] is
  A)
  2 sq. units done clear
  B)
  4 sq. units done clear
  C)
  6 sq. units done clear
  D)
  8 sq. units done clear
  View Solution play_arrow

• question_answer2)
  If the angle between the two lines represented by \[2{{x}^{2}}+5xy+3{{y}^{2}}+6x+7y+4=0\] is \[{{\tan }^{-1}}m.\] then m is equal to:
  A)
  \[\frac{1}{5}\] done clear
  B)
  1 done clear
  C)
  \[\frac{7}{5}\] done clear
  D)
  7 done clear
  View Solution play_arrow

• question_answer3)
  Locus of midpoint of the portion between the axes of \[x\cos \alpha +y\sin \alpha =p\] where p is constant is
  A)
  \[{{x}^{2}}+{{y}^{2}}=\frac{4}{{{p}^{2}}}\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}=4{{p}^{2}}\] done clear
  C)
  \[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}=\frac{2}{{{p}^{2}}}\] done clear
  D)
  \[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}=\frac{4}{{{p}^{2}}}\]   done clear
  View Solution play_arrow

• question_answer4)
  A straight line cuts off an intercept of 2 units on the positive direction of x-axis and passes through the point (-3, 5). What is the foot of the perpendicular drawn from the point (3, 3) on this line?
  A)
  \[(1,3)\] done clear
  B)
  \[(2,0)\] done clear
  C)
  \[(0,2)\] done clear
  D)
  \[(1,1)\] done clear
  View Solution play_arrow

• question_answer5)
  Let \[0<\alpha <\pi /2\] be a fixed angle. If \[P(cos\theta ,sin\theta )\] and \[Q(cos(\alpha -\theta ),sin(\alpha -\theta )),\] then Q is obtained from P by the
  A)
  Clockwise rotation around the origin through an angle \[\alpha \] done clear
  B)
  Anticlockwise rotation around the origin through an angle \[\alpha \] done clear
  C)
  Reflection in the line through the origin with slope \[\tan \alpha \] done clear
  D)
  Reflection in the line through the origin with slop \[\tan (\alpha /2)\] done clear
  View Solution play_arrow

• question_answer6)
  The combined equation of the pair of lines through the point (1, 0) and parallel to the lines represented by \[2{{x}^{2}}-xy-{{y}^{2}}=0\] is
  A)
  \[2{{x}^{2}}-xy-{{y}^{2}}-4x-y=0\] done clear
  B)
  \[2{{x}^{2}}-xy-{{y}^{2}}-4x+y+2=0\] done clear
  C)
  \[2{{x}^{2}}+xy+{{y}^{2}}-2x+y=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  If the slope of one of the lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] is the square of the other, Then \[\frac{a+b}{h}+\frac{8{{h}^{2}}}{ab}=\]
  A)
  4 done clear
  B)
  6 done clear
  C)
  8 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer8)
  The indenter of a triangle with vertices (7, 1), (-1, 5) and \[(3+2\sqrt{3},\,\,3+4\sqrt{3})\] is
  A)
  \[\left( 3+\frac{2}{\sqrt{3}},3+\frac{4}{\sqrt{3}} \right)\] done clear
  B)
  \[\left( 1+\frac{2}{3\sqrt{3}},1+\frac{4}{3\sqrt{3}} \right)\] done clear
  C)
  \[(7,1)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  What is the radius of the circle passing through the point \[(2,4)\] and having centre at the intersection of the lines \[x-y=4\] and \[2x+3y+7=0?\]
  A)
  3 units done clear
  B)
  5 units done clear
  C)
  \[3\sqrt{3}\] Units done clear
  D)
  \[5\sqrt{2}\] units done clear
  View Solution play_arrow

• question_answer10)
  If 2p is the length of perpendicular from the origin to the lines \[\frac{x}{a}+\frac{y}{b}=1\], then \[{{a}^{2}},8{{p}^{2}},{{b}^{2}}\] 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_answer11)
  If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is
  A)
  Square done clear
  B)
  Circle done clear
  C)
  Straight line done clear
  D)
  Two intersecting line done clear
  View Solution play_arrow

• question_answer12)
  The line \[x+3y-2=0\]bisects the angle between a pair of straight lines of which one has equation\[x-7y+5=0\]. The equation of the other line is
  A)
  \[3x+3y-1=0\] done clear
  B)
  \[x-3y+2=0\] done clear
  C)
  \[5x+5y-3=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  The equation \[8{{x}^{2}}+8xy+2{{y}^{2}}+26x+13y+15=0\] represents a pair of straight lines. The distance between them is
  A)
  \[7/\sqrt{5}\] done clear
  B)
  \[7/2\sqrt{5}\] done clear
  C)
  \[\sqrt{7}/5\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer14)
  A line L intersects the three sides BC. CA and AB of a\[\Delta ABC\]at P, Q and R respectively. Then, \[\frac{BP}{PC}.\frac{CQ}{QA}.\frac{AR}{RB}\] is equal to
  A)
  1 done clear
  B)
  0 done clear
  C)
  -1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer15)
  What is the acute angle between the lines represented by the equations \[y-\sqrt{3}x-5=0\] and \[\sqrt{3}x-x+6=0?\]
  A)
  \[30{}^\circ \] done clear
  B)
  \[45{}^\circ \] done clear
  C)
  \[60{}^\circ \] done clear
  D)
  \[75{}^\circ \] done clear
  View Solution play_arrow

• question_answer16)
  A regular polygon with equal sides has 9 diagonals. Two of the vertices are at \[A(-1,0)\] and\[B(1,0)\]. Possible areas of polygon is
  A)
  \[\frac{3\sqrt{3}}{2},2\sqrt{3},6\sqrt{3}\] done clear
  B)
  \[2\sqrt{3},3\sqrt{3},6\sqrt{3}\] done clear
  C)
  \[9\sqrt{3},6\sqrt{3},2\sqrt{3}\] done clear
  D)
  \[\frac{3\sqrt{3}}{2},3\sqrt{3},6\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer17)
  A line which makes an acute angle \[\theta \] with the positive direction of x-axis is drawn through the point P(3, 4) to meet the line \[x=6\] at R and \[y=8\] at S, then
  A)
  \[PR=3\cos \theta \]        done clear
  B)
  \[PS=-4\cos ec\theta \] done clear
  C)
  \[PR-PS=\frac{2(3sin\theta +4cos\theta )}{\sin 2\theta }\] done clear
  D)
  \[\frac{9}{{{(PR)}^{2}}}+\frac{16}{{{(PS)}^{2}}}=1\] done clear
  View Solution play_arrow

• question_answer18)
  Locus of centroid of the triangle whose vertices are \[(\alpha cos\,t,a\,sin\,t),(b\,sin\,t,-b\,cos\,t)\] and \[(1,0)\], where t is a parameter, is
  A)
  \[{{(3x+1)}^{2}}+{{(3y)}^{2}}={{a}^{2}}-{{b}^{2}}\] done clear
  B)
  \[{{(3x-1)}^{2}}+{{(3y)}^{2}}={{a}^{2}}-{{b}^{2}}\] done clear
  C)
  \[{{(3x-1)}^{2}}+{{(3y)}^{2}}={{a}^{2}}+{{b}^{2}}\] done clear
  D)
  \[{{(3x+1)}^{2}}+{{(3y)}^{2}}={{a}^{2}}+{{b}^{2}}\] done clear
  View Solution play_arrow

• question_answer19)
  The range of value of \[\alpha \] such that \[(0,\alpha )\] lies on or inside the triangle formed by the lines \[y+3x+2=0,\] \[3y-2x-5=0,\] \[4y+x-14=0\]is
  A)
  \[5<\alpha \le 7\] done clear
  B)
  \[\frac{1}{2}\le \alpha \le 1\] done clear
  C)
  \[\frac{5}{3}\le \alpha \le \frac{7}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  If \[{{p}_{1}},{{p}_{2}}\] are the lengths of the normal drawn from the origin on the lines\[x\cos \theta +ysin\theta =2acos4\theta \] And \[x\sec \theta +y\cos ec\theta =4a\cos 2\theta \] respectively, and \[m{{p}^{2}}_{1}+n{{p}^{2}}_{2}=4{{a}^{2}}.\] Then
  A)
  \[m=1,n=1\] done clear
  B)
  \[m=1,n=4\] done clear
  C)
  \[m=4,n=1\] done clear
  D)
  \[m=1,n=-1\] done clear
  View Solution play_arrow

• question_answer21)
  The area of the figure formed by the lines \[ax+by+c=0,ax-by+c=0,ax+by-c=0\]and \[ax-by-c=0\] is
  A)
  \[\frac{{{c}^{2}}}{ab}\] done clear
  B)
  \[\frac{2{{c}^{2}}}{ab}\] done clear
  C)
  \[\frac{{{c}^{2}}}{2ab}\] done clear
  D)
  \[\frac{{{c}^{2}}}{4ab}\] done clear
  View Solution play_arrow

• question_answer22)
  Two straight lines passing through the point \[A(3,2)\] cut the line \[2y=x+3\] and x-axis perpendicularly at P and Q respectively. The equation of the line PQ is
  A)
  \[7x+y-21=0\] done clear
  B)
  \[x+7y+21=0\] done clear
  C)
  \[2x+y-8=0\] done clear
  D)
  \[x+2y+8=0\] done clear
  View Solution play_arrow

• question_answer23)
  Let \[(h,k)\] be a fixed point where \[h>0,k>0.\] A straight line passing through this point cuts the positive direction of the coordinate axes at the points P and Q. Then the minimum area of the \[\Delta OPQ.O\] O being the origin, is
  A)
  4hk sq. units done clear
  B)
  2hk sq. units done clear
  C)
  3hk sq. units done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer24)
  The line \[{{L}_{1}}:4x+3y-12=0\]intersects the x-and y-axis at A and B, respectively. A variable line perpendicular to \[{{L}_{1}}\] intersects the x- and the y. the circumventer of triangle ABQ is
  A)
  \[3x-4y+2=0\] done clear
  B)
  \[4x+3y+7=0\] done clear
  C)
  \[6x-8y+7=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer25)
  If the point \[P(x,y)\] is equidistant from points\[A(a+b,b-a)\] and \[B(a-b,a+b)\], then
  A)
  \[ax=by\] done clear
  B)
  \[bx=ay\] and P ca be (a, b) done clear
  C)
  \[{{x}^{2}}-{{y}^{2}}=2(ax+by)\] done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer26)
  Let \[A\left( \alpha ,\frac{1}{\alpha } \right),B\left( \alpha ,\frac{1}{\beta } \right),C\left( \gamma ,\frac{1}{\gamma } \right)\] be the vertices of a \[\Delta ABC,\] where \[\alpha ,\beta \] are the roots of the equation \[{{x}^{2}}-6{{p}_{1}}x+2=0,\,\,\beta ,\,\,\gamma \]\[{{x}^{2}}-6{{p}_{1}}x+2=0,\beta ,\gamma \] are the roots of the equation \[{{x}^{2}}-6{{p}_{2}}x+3=0\] and \[\gamma ,\alpha \] are the roots of the equation \[{{x}^{2}}-6{{p}_{3}}x+6=0,{{p}_{1}},{{p}_{2}},{{p}_{3}}\] being positive. Then, the coordinates of the centroid of \[\Delta ABC\] is
  A)
  \[\left( 1,\frac{11}{18} \right)\] done clear
  B)
  \[\left( 0,\frac{11}{8} \right)\] done clear
  C)
  \[\left( 2,\frac{11}{18} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer27)
  The bisector of the acute angle formed between the lines \[4x-3y+7=0\] and \[3x-4y+14=0\] has the equation:
  A)
  \[x+y+3=0\] done clear
  B)
  \[x-y-3=0\] done clear
  C)
  \[x-y+3=0\] done clear
  D)
  \[3x+y-7=0\] done clear
  View Solution play_arrow

• question_answer28)
  The equation \[{{({{x}^{2}}-{{a}^{2}})}^{2}}{{({{x}^{2}}-{{b}^{2}})}^{2}}+{{c}^{4}}{{({{y}^{2}}-{{a}^{2}})}^{2}}=0\] represents \[(c\ne 0)\]
  A)
  8 points done clear
  B)
  Two circles done clear
  C)
  4 lines done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer29)
  The line \[x+y=a\] meets the axes of x and y at A and B respectively. A \[\Delta AMN\] is inscribed in the \[\Delta OAB,\,\,O\] being the origin, with right angle at N. M and N lie respectively on OB and AB. If the area of the \[\Delta AMN\] is \[\frac{3}{8}\] of the area of the \[\Delta OAB,\] then \[\frac{AN}{BN}\] is equal to
  A)
  \[\frac{1}{3}\] done clear
  B)
  \[\frac{1}{3},3\] done clear
  C)
  \[\frac{2}{3},3\] done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer30)
  The lines \[2x=3y=-z\] and \[6x=-y=-4z\]
  A)
  Are perpendicular done clear
  B)
  Are parallel done clear
  C)
  \[intersect\text{ }at\text{ }an\text{ }angle\,\,45{}^\circ \] done clear
  D)
  \[intersect\text{ }at\text{ }an\text{ }angle\,\,60{}^\circ \] done clear
  View Solution play_arrow

• question_answer31)
  Given a family of lines a \[a(2x+y+4)+b(x-2y-3)=0\] the number of lines belonging to the family at a distance \[\sqrt{10}\] from \[P(2,-3)\] is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer32)
  If the line segment joining the points \[A(a,b)\] and \[B(c,d)\] subtends an angle \[\theta \] at the origin, then \[\cos \theta =\]
  A)
  \[\frac{ac+bd}{\sqrt{({{a}^{2}}+{{b}^{2}})({{c}^{2}}+{{d}^{2}})}}\] done clear
  B)
  \[\frac{ab+cd}{\sqrt{({{a}^{2}}+{{b}^{2}})({{c}^{2}}+{{d}^{2}})}}\] done clear
  C)
  \[\frac{ad+bc}{\sqrt{({{a}^{2}}+{{b}^{2}})({{c}^{2}}+{{d}^{2}})}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer33)
  The straight lines \[x+2y-9=0,\,\,\,3x+5y-5=0\] and \[ax+by=1\] are concurrent if the straight line \[35x-22y+1=0\] passes through:
  A)
  (a, b) (b) done clear
  B)
  (b, a) done clear
  C)
  (a, -b) (d) done clear
  D)
  (-a, b) done clear
  View Solution play_arrow

• question_answer34)
  The equation of the straight line which passes through the point \[(-4,3)\] such that the portion of the line between the axes is divided internally by the point in the ration 5 : 3 is
  A)
  \[9x-20y+96=0\] done clear
  B)
  \[9x+20y=24\] done clear
  C)
  \[20x+9y+53=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer35)
  D is a point on AC of the triangle with vertices A \[(2,3)\], \[B(1,-3)\], \[C(-4,-7)\] and BD divides ABC into two triangles of equal area. The equation of the line drawn though B at right angles to BD is
  A)
  \[y-2x+5=0\] done clear
  B)
  \[2y-x+5=0\] done clear
  C)
  \[y+2x-5=0\] done clear
  D)
  \[2y+x-5=0\] done clear
  View Solution play_arrow

• question_answer36)
  Vertices of a variable triangle are \[(3,4),\]\[(5cos\theta ,5sin\theta )\] and \[(5sin\theta ,-5cos\theta ),\]where \[\theta \in R.\] Locus of its orthocenter is
  A)
  \[{{(x+y-1)}^{2}}+{{(x-y-7)}^{2}}=100\] done clear
  B)
  \[{{(x+y-7)}^{2}}+{{(x-y-1)}^{2}}=100\] done clear
  C)
  \[{{(x+y-7)}^{2}}+{{(x+y-1)}^{2}}=100\] done clear
  D)
  \[{{(x+y-7)}^{2}}+{{(x-y+1)}^{2}}=100\] done clear
  View Solution play_arrow

• question_answer37)
  The length of the perpendicular from the origin to a line is 7 and line makes an angle of \[150{}^\circ \] with the positive direction of y-axis, then the equation of the line is
  A)
  \[\sqrt{3}x+y=7\] done clear
  B)
  \[\sqrt{3}x-y=14\] done clear
  C)
  \[\sqrt{3}x+y+14=0\] done clear
  D)
  \[\sqrt{3}x+y-14=0\] done clear
  View Solution play_arrow

• question_answer38)
  The number of equilateral triangles with \[y=\sqrt{3}(x-1)+2\] and \[y=-\sqrt{3x}\] as two of its sides is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer39)
  What is the equation of the line through \[(1,2)\] so that segment of the line intercepted between the axes is bisected at this point?
  A)
  \[2x-y=4\] done clear
  B)
  \[2x-y+4=0\] done clear
  C)
  \[2x+y=4\] done clear
  D)
  2x+y+4=0 done clear
  View Solution play_arrow

• question_answer40)
  If the straight lines \[ax+may+1=0,\] \[bx+(m+1)by+1=0\] and \[cx+(m+2)cy+1=0\] are concurrent, then a, b, c form \[(m\ne 0)\]
  A)
  An A.P. only for m=1 done clear
  B)
  An A.P. for all m done clear
  C)
  A G.P. for all m done clear
  D)
  A H.P. for all m done clear
  View Solution play_arrow

• question_answer41)
  The diagonals of the parallelogram whose sides are \[lx+my+n=0,\] \[lx+my+n'=0\], \[mx+1y+n=0\] and \[mx+ly+n'=0\] include an angle
  A)
  \[\frac{\pi }{3}\] done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  \[{{\tan }^{-1}}\left( \frac{{{1}^{2}}-{{m}^{2}}}{{{1}^{2}}+{{m}^{2}}} \right)\] done clear
  D)
  \[{{\tan }^{-1}}\left( \frac{2lm}{{{1}^{2}}+{{m}^{2}}} \right)\] done clear
  View Solution play_arrow

• question_answer42)
  A light ray emerging from the point source placed at P (2, 3) is reflected at a point Q on the y-axis. It then passes through the point \[R(5,10).\] The coordinates of Q are
  A)
  \[(0,3)\] done clear
  B)
  \[(0,2)\] done clear
  C)
  \[(0,5)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer43)
  Through the point \[P(\alpha ,\beta )\], where \[\alpha \beta >0.\] the straight line \[\frac{x}{a}+\frac{y}{b}=1\] is drawn so as the form with axes a triangle of area S. if \[ab>0,\] then least value of S is
  A)
  \[\alpha \beta \] done clear
  B)
  \[2\alpha \beta \] done clear
  C)
  \[3\alpha \beta \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer44)
  P is point on the line \[y+2x=1,\] and Q and R are two points on the line \[3y+6x=6\]such that triangle PQR is an equilateral triangle. The length of the side of the triangle is
  A)
  \[2/\sqrt{15}\] done clear
  B)
  \[3/\sqrt{5}\] done clear
  C)
  \[4/\sqrt{5}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer45)
  What is the angle between the lines \[x+y=1\]and \[x-y=1?\]
  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_answer46)
  A straight line L with negative slope passes through the point (8, 2) and cuts the positive coordinate axes at points P and Q. as L varies the absolute minimum value of \[OP+OQ\] is (O is origin)
  A)
  28 done clear
  B)
  15 done clear
  C)
  18 done clear
  D)
  10 done clear
  View Solution play_arrow

• question_answer47)
  Consider the points \[A(0,1)\] and \[B(2,0),\] and P be a point on the line \[4x+3y+9=0.\] The coordinates of P such that \[\left| PA-PB \right|\] is maximum are
  A)
  \[(-12/5,17/5)\] done clear
  B)
  \[(-84/5,13/5)\] done clear
  C)
  \[(-6/5,17/5)\] done clear
  D)
  \[(0,-3)\] done clear
  View Solution play_arrow

• question_answer48)
  The point \[A(2,1)\] is translated parallel to the line \[x-y=3\] by, a distance of 4 units. If the new position A? is in the third quadrant, then the coordinates of A? are
  A)
  \[(2+2\sqrt{2},1+2\sqrt{2})\] done clear
  B)
  \[(-2+\sqrt{2},-1-2\sqrt{2})\] done clear
  C)
  \[(2-2\sqrt{2},1-2\sqrt{2})\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer49)
  A ray of light passing through a point (1, 2) is reflected on the x-axis at point Q and passes through the point (5, 8). Then the abscissa of the point Q is
  A)
  -3 done clear
  B)
  9/5 done clear
  C)
  13/5      done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer50)
  The equation of straight line passing through (-a, 0) and making a triangle with the axes of area T is
  A)
  \[2Tx+{{a}^{2}}y+2aT=0\] done clear
  B)
  \[2Tx-{{a}^{2}}y+2aT=0\] done clear
  C)
  \[2Tx-{{a}^{2}}y-2aT=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer51)
  What is the equation of the line which passes through (4, -5) and is perpendicular to\[3x+4y+5=0?\]
  A)
  \[4x-3y-31=0\] done clear
  B)
  \[3x-4y-41=0\] done clear
  C)
  \[4x+3y-1=0\] done clear
  D)
  \[3x+4y+8=0\] done clear
  View Solution play_arrow

• question_answer52)
  Suppose A, B are two points on \[2x-y+3=0\] and P (1, 2) is such that \[PA=PB.\] Then the mid-point of AB is
  A)
  \[\left( -\frac{1}{5},\frac{13}{5} \right)\] done clear
  B)
  \[\left( \frac{-7}{5},\frac{9}{5} \right)\] done clear
  C)
  \[\left( \frac{7}{5},\frac{-9}{5} \right)\] done clear
  D)
  \[\left( \frac{-7}{5},\frac{-9}{5} \right)\] done clear
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• question_answer53)
  P (m, n) (Where m, n are natural numbers) is an point in the interior of the quadrilateral formed by the pair of lines xy=0 and the two lines \[2x+y-2=0\] and \[4x+5y=20.\] The possible number of positions of the point P is
  A)
  Six done clear
  B)
  Five done clear
  C)
  Four      done clear
  D)
  Eleven   done clear
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• question_answer54)
  A variable line ?L? is drawn through \[O(0,\,\,0)\] to meet the lines \[{{L}_{1}}:y-x-10=0\] and \[{{L}_{2}}:y-x-20=0\] at the points A and B respectively. A point P is taken on ?L? such that\[\frac{2}{OP}=\frac{1}{OA}+\frac{1}{OB}.\] Locus of ?P? is
  A)
  \[3x+3y=40\] done clear
  B)
  \[3x+3y+40=0\] done clear
  C)
  \[3x-3y=40\]      done clear
  D)
  \[3y-3x=40\] done clear
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• question_answer55)
  The middle point of the segment of the straight line joining the points (p, q) and (q, -p) is \[(r/2),s/2\] what is the length of the segment?
  A)
  \[[{{({{s}^{2}}+{{r}^{2}})}^{1/2}}]/2\] done clear
  B)
  \[[{{({{s}^{2}}+{{r}^{2}})}^{1/2}}]/4\] done clear
  C)
  \[{{({{s}^{2}}+{{r}^{2}})}^{1/2}}\] done clear
  D)
  \[s+r\] done clear
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• question_answer56)
  If the sum of the squares of the distances of the point (x, y) from the pints (a, 0) and (-a, 0) is \[2{{b}^{2}},\] then which one of the following is correct?
  A)
  \[{{x}^{2}}+{{a}^{2}}={{b}^{2}}+{{y}^{2}}\] done clear
  B)
  \[{{x}^{2}}+{{a}^{2}}=2{{b}^{2}}-{{y}^{2}}\] done clear
  C)
  \[{{x}^{2}}-{{a}^{2}}={{b}^{2}}+{{y}^{2}}\] done clear
  D)
  \[{{x}^{2}}+{{a}^{2}}={{b}^{2}}-{{y}^{2}}\] done clear
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• question_answer57)
  A rectangle ABCD, where A(0, 0), B(4, 0), C(4, 2), D(0, 2), undergoes the following transformations successively:

  i. \[{{f}_{1}}(x,y)\to (y,x)\]

  ii. \[{{f}_{2}}(x,y)\to (x+3y,y)\]

  iii. \[{{f}_{3}}(x,y)\to ((x-y)/2,(x+y)/2)\]

  The final figure will be
  A)
  A square done clear
  B)
  A rhombus done clear
  C)
  A rectangle done clear
  D)
  A parallelogram done clear
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• question_answer58)
  A straight line through the origin O meets the parallel lines \[4x+2y=9\] and \[2x+y+6=0\] at points P and Q respectively. Then the points O divides the segment PQ in the ratio
  A)
  1:2 done clear
  B)
  3:4 done clear
  C)
  2:1 done clear
  D)
  4:3 done clear
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• question_answer59)
  For \[a>b>c>0,\] the distance between (1, 1) and the point of intersection of the lines \[ax+by+c=0\] and \[bx+ay+c=0\] is less than \[2\sqrt{2}.\] Then
  A)
  \[a+b-c>0\] done clear
  B)
  \[a-b+c<0\] done clear
  C)
  \[a-b+c>0\] done clear
  D)
  \[a+b-c<0\] done clear
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• question_answer60)
  If \[(-4,5)\] is one vertex and \[7x-y+8=0\] is one diagonal of a square, then he equation of second diagonal is
  A)
  \[x+3y=21\] done clear
  B)
  \[2x-3y=7\] done clear
  C)
  \[x+7y=31\] done clear
  D)
  \[2x+3y=21\] done clear
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• question_answer61)
  Two points \[P(a,0)\] and \[Q(-a,0)\] are given, R is a variable point on one side of the line PQ such that \[\angle RPQ-\angle RQP\] is \[2\alpha \]. Then, the locus of R is
  A)
  \[{{x}^{2}}-{{y}^{2}}+2xy\,\,\cot \,\,2\alpha -{{a}^{2}}=0\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}+2xy\,\,\cot \,\,2\alpha -{{a}^{2}}=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}+2xy\,\,\cot \,\,2\alpha +{{a}^{2}}=0\] done clear
  D)
  None of the above done clear
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• question_answer62)
  The circumradius of the triangle formed by the three lines \[y+3x-5=0;y=x\] and \[3y-x+10=0\] is
  A)
  \[\frac{25}{4\sqrt{2}}\] done clear
  B)
  \[\frac{25}{3\sqrt{2}}\] done clear
  C)
  \[\frac{25}{2\sqrt{2}}\] done clear
  D)
  \[\frac{25}{\sqrt{2}}\] done clear
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• question_answer63)
  The intercept cut off by a line from y-axis twice than that form x-axis, and the line passes through the point (1, 2). The equation of the line is
  A)
  \[2x+y=4\] done clear
  B)
  \[2x+y+4=0\] done clear
  C)
  \[2x-y=4\] done clear
  D)
  \[2x-y+4=0\] done clear
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• question_answer64)
  Points \[P(p,0),Q(q,0),R(0,p),S(0,q)\] form
  A)
  Parallelogram done clear
  B)
  Rhombus done clear
  C)
  Cyclic quadrilateral done clear
  D)
  None of these done clear
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• question_answer65)
  If the points (h, 0), (a, b) and (o, k) lies on a line, then the value of \[\frac{a}{h}+\frac{b}{k}\] is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  3 done clear
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• question_answer66)
  From the point (4, 3) a perpendicular is dropped on the x-axis as well as on the y-axis. If the lengths of perpendiculars are p, q respectively, then which one of the following is correct?
  A)
  \[p=q~\] done clear
  B)
  \[3p=4q\] done clear
  C)
  \[4p=3q\] done clear
  D)
  \[p+q=5\]   done clear
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• question_answer67)
  The area of the region bounded by the locus of a point P satisfying \[d(P,A)=4\], where A is (1, 2) is
  A)
  64 sq. unit done clear
  B)
  54 sq. unit done clear
  C)
  \[\,16\pi \,\,sq.\text{ }unit\] done clear
  D)
  None of these done clear
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• question_answer68)
  The point \[({{t}^{2}}+2t+5,2{{t}^{2}}+t-2)\] lies on the line\[x+y=2\] for
  A)
  All real values of t done clear
  B)
  Some real values of t done clear
  C)
  \[t=\frac{-3\pm \sqrt{3}}{6}\] done clear
  D)
  None of these done clear
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• question_answer69)
  Let \[P=(-1,0),Q=(0,0)\] and \[R=(3,3\sqrt{3})\] be three point. The equation of the bisector of the angle PQR is
  A)
  \[\frac{\sqrt{3}}{2}x+y=0\] done clear
  B)
  \[x+\sqrt{3y}=0\] done clear
  C)
  \[\sqrt{3}x+y=0\] done clear
  D)
  \[x+\frac{\sqrt{3}}{2}y=0\] done clear
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• question_answer70)
  If the lines \[y=(2+\sqrt{3})x+4\] and \[y=kx+6\] are inclined at an angle \[60{}^\circ \] to each other, then the value of k will be
  A)
  1 done clear
  B)
  2 done clear
  C)
  -1 done clear
  D)
  -2 done clear
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