JEE Main & Advanced Mathematics Rectangular Cartesian Coordinates Question Bank

done Transformation of axes and Locus Total Questions - 40

• question_answer1)
  The new coordinates of a point (4, 5), when the origin is shifted to the point (1,-2) are [MNR 1988; IIT 1989; UPSEAT 2000]
  A)
  (5, 3) done clear
  B)
  (3, 5) done clear
  C)
  (3, 7) done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  Without changing the direction of coordinate axes, origin is transferred to \[(h,k)\], so that the linear (one degree) terms in the equation \[{{x}^{2}}+{{y}^{2}}-4x+6y-7\]=0 are eliminated. Then the point \[(h,k)\]is
  A)
  (3, 2) done clear
  B)
  (- 3, 2) done clear
  C)
  (2, - 3) done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  The equation of the locus of a point whose distance from (a, 0) is equal to its distance from y-axis, is [MP PET 1986]
  A)
  \[{{y}^{2}}-2ax={{a}^{2}}\] done clear
  B)
  \[{{y}^{2}}-2ax+{{a}^{2}}=0\] done clear
  C)
  \[{{y}^{2}}+2ax+{{a}^{2}}=0\] done clear
  D)
  \[{{y}^{2}}+2ax={{a}^{2}}\] done clear
  View Solution play_arrow

• question_answer4)
  Two points A and B have coordinates (1, 0) and (-1, 0) respectively and Q is a point which satisfies the relation \[AQ-BQ=\]\[\pm 1.\]The locus of Q is [MP PET 1986]
  A)
  \[12{{x}^{2}}+4{{y}^{2}}=3\] done clear
  B)
  \[12{{x}^{2}}-4{{y}^{2}}=3\] done clear
  C)
  \[12{{x}^{2}}-4{{y}^{2}}+3=0\] done clear
  D)
  \[12{{x}^{2}}+4{{y}^{2}}+3=0\] done clear
  View Solution play_arrow

• question_answer5)
  The locus of a point P which moves in such a way that the segment OP, where O is the origin, has slope \[\sqrt{3}\] is
  A)
  \[x-\sqrt{3}y=0\] done clear
  B)
  \[x+\sqrt{3}y=0\] done clear
  C)
  \[\sqrt{3}x+y=0\] done clear
  D)
  \[\sqrt{3}x-y=0\] done clear
  View Solution play_arrow

• question_answer6)
  If the coordinates of a point be given by the equation \[x=a(1-\cos \theta ),\]\[y=a\sin \theta \], then the locus of the point will be
  A)
  A straight line done clear
  B)
  A circle done clear
  C)
  A parabola done clear
  D)
  An ellipse done clear
  View Solution play_arrow

• question_answer7)
  If P = (1,0), Q =(-1,0) and R =(2,0) are three given points, then the locus of a point S satisfying the relation \[S{{Q}^{2}}+S{{R}^{2}}=2S{{P}^{2}}\] is [IIT 1988]
  A)
  A straight line parallel to x-axis done clear
  B)
  A circle through origin done clear
  C)
  A circle with centre at the origin done clear
  D)
  A straight line parallel to y-axis done clear
  View Solution play_arrow

• question_answer8)
  The coordinates of the points O, A and B are (0,0), (0,4) and (6,0) respectively. If a points P moves such that the area of \[\Delta POA\]is always twice the area of \[\Delta POB\], then the equation to both parts of the locus of P is [IIT 1964]
  A)
  \[(x-3y)(x+3y)=0\] done clear
  B)
  \[(x-3y)(x+y)=0\] done clear
  C)
  \[(3x-y)(3x+y)=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  A point moves in such a way that the sum of square of its distance from the points \[A(2,0)\]and\[B(-2,0)\]is always equal to the square of the distance between A and B. The locus of the point is
  A)
  \[{{x}^{2}}+{{y}^{2}}-2=0\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}+2=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}+4=0\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}-4=0\] done clear
  View Solution play_arrow

• question_answer10)
  A point P moves so that its distance from the point \[(a,0)\]is always equal to its distance from the line \[x+a=0\]. The locus of the point is [MP PET 1982]
  A)
  \[{{y}^{2}}=4ax\] done clear
  B)
  \[{{x}^{2}}=4ay\] done clear
  C)
  \[{{y}^{2}}+4ax=0\] done clear
  D)
  \[{{x}^{2}}+4ay=0\] done clear
  View Solution play_arrow

• question_answer11)
  The equation to the locus of a point which moves so that its distance from x-axis is always one half its distance from the origin, is  
  A)
  \[{{x}^{2}}+3{{y}^{2}}=0\] done clear
  B)
  \[{{x}^{2}}-3{{y}^{2}}=0\] done clear
  C)
  \[3{{x}^{2}}+{{y}^{2}}=0\] done clear
  D)
  \[3{{x}^{2}}-{{y}^{2}}=0\] done clear
  View Solution play_arrow

• question_answer12)
  A point moves so that its distance from the point (-1, 0) is always three times its distance from the point (0, 2). The locus of the point is   
  A)
  A line done clear
  B)
  A circle done clear
  C)
  A parabola done clear
  D)
  An ellipse done clear
  View Solution play_arrow

• question_answer13)
  The locus of a point which moves so that its distance from  x-axis is double of its distance from y-axis is [AMU 1978; MP PET 1984]
  A)
  \[x=2y\] done clear
  B)
  \[y=2x\] done clear
  C)
  \[x=5y+1\] done clear
  D)
  \[y=2x+3\] done clear
  View Solution play_arrow

• question_answer14)
  O is the origin and A is the point (3, 4). If a point P moves so that the line segment OP is always parallel to the line segment OA, then the equation to the locus of P is  
  A)
  \[4x-3y=0\] done clear
  B)
  \[4x+3y=0\] done clear
  C)
  \[3x+4y=0\] done clear
  D)
  \[3x-4y=0\] done clear
  View Solution play_arrow

• question_answer15)
  The locus of a point which moves so that it is always equidistant from the point A(a, 0) and B (- a, 0) is  [MP PET 1984]
  A)
  A circle done clear
  B)
  Perpendicular bisector of the line segment AB done clear
  C)
  A line parallel to x-axis done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer16)
  The coordinates of the points A and B are (a, 0) and \[(-a,\,0)\] respectively. If a point P moves so that \[P{{A}^{2}}-P{{B}^{2}}=2{{k}^{2}}\], when k is constant, then the equation to the locus of the point P , is   
  A)
  \[2ax-{{k}^{2}}=0\] done clear
  B)
  \[2ax+{{k}^{2}}=0\] done clear
  C)
  \[2ay-{{k}^{2}}=0\] done clear
  D)
  \[2ay+{{k}^{2}}=0\] done clear
  View Solution play_arrow

• question_answer17)
  If the coordinates of a point be given by the equations \[x=b\sec \varphi ,\ \ y=a\tan \varphi \], then its locus is
  A)
  A straight line done clear
  B)
  A circle done clear
  C)
  An ellipse done clear
  D)
  A hyperbola done clear
  View Solution play_arrow

• question_answer18)
  The coordinates of the point A and B are \[(ak,0)\] and\[\left( \frac{a}{k},0 \right),\,\,(k=\pm 1)\]. If a point P moves so that \[PA=kPB,\] then the equation to the locus of P is
  A)
  \[{{k}^{2}}({{x}^{2}}+{{y}^{2}})-{{a}^{2}}=0\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}-{{k}^{2}}{{a}^{2}}=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}+{{a}^{2}}=0\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}-{{a}^{2}}=0\] done clear
  View Solution play_arrow

• question_answer19)
  The locus of a point which moves in such a way that its distance from (0,0) is three times its distance from the x-axis, as given by [MP PET 1993]
  A)
  \[{{x}^{2}}-8{{y}^{2}}=0\] done clear
  B)
  \[{{x}^{2}}+8{{y}^{2}}=0\] done clear
  C)
  \[4{{x}^{2}}-{{y}^{2}}=0\] done clear
  D)
  \[{{x}^{2}}-4{{y}^{2}}=0\] done clear
  View Solution play_arrow

• question_answer20)
  The equation of the locus of all points equidistant from the point (4,2) and the x-axis, is     [CEE 1993]
  A)
  \[{{x}^{2}}+8x+4y-20=0\] done clear
  B)
  \[{{x}^{2}}-8x-4y+20=0\] done clear
  C)
  \[{{y}^{2}}-4y-8x+20=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer21)
  The locus of the mid-point of the distance between the axes of the variable line \[x\cos \alpha +y\sin \alpha =p,\]where p is constant,  is   [MNR 1985; CEE 1993; UPSEAT 2000; AIEEE 2002]
  A)
  \[{{x}^{2}}+{{y}^{2}}=4{{p}^{2}}\] done clear
  B)
  \[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}=\frac{4}{{{p}^{2}}}\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}=\frac{4}{{{p}^{2}}}\] done clear
  D)
  \[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}=\frac{2}{{{p}^{2}}}\] done clear
  View Solution play_arrow

• question_answer22)
  The locus of a point whose distance from the point \[(-g,-f)\]is always 'a', will be, (where \[k={{g}^{2}}+{{f}^{2}}-{{a}^{2}}\])  
  A)
  \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+k=0\] done clear
  B)
  \[{{x}^{2}}-{{y}^{2}}+2gx+2fy+k=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}+2xy+2gx+2fy+k=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer23)
  The locus of the moving point P, such that 2PA = 3PB where A is (0,0) and B is (4,-3), is [AMU 1980]
  A)
  \[5{{x}^{2}}-5{{y}^{2}}-72x+54y+225=0\] done clear
  B)
  \[5{{x}^{2}}-5{{y}^{2}}+72x+54y+225=0\] done clear
  C)
  \[5{{x}^{2}}+5{{y}^{2}}+72x+54y+225=0\] done clear
  D)
  \[5{{x}^{2}}+5{{y}^{2}}-72x+54y+225=0\] done clear
  View Solution play_arrow

• question_answer24)
  A point moves such that the sum of its distances from two fixed points (ae,0) and (-ae,0) is always 2a. Then equation of its locus is [MNR 1981]
  A)
  \[\]\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}(1-{{e}^{2}})}=1\] done clear
  B)
  \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{a}^{2}}(1-{{e}^{2}})}=1\] done clear
  C)
  \[\frac{{{x}^{2}}}{{{a}^{2}}(1-{{e}^{2}})}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer25)
  A point moves in such a way that its distance from (1,-2) is always the twice from (-3,5), the locus of the point is  
  A)
  \[3{{x}^{2}}+{{y}^{2}}+26x+44y-131=0\] done clear
  B)
  \[{{x}^{2}}+3{{y}^{2}}-26x+44y-131=0\] done clear
  C)
  \[3({{x}^{2}}+{{y}^{2}})+26x-44y+131=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer26)
  A point moves in such a way that its distance from origin is always 4. Then the locus of the point is       
  A)
  \[{{x}^{2}}+{{y}^{2}}=4\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}=16\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}=2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer27)
  If \[A(-a,0)\] and \[B(a,0)\]are two fixed points, then the locus of the point on which the line AB subtends the right angle, is 
  A)
  \[{{x}^{2}}+{{y}^{2}}=2{{a}^{2}}\] done clear
  B)
  \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}+{{a}^{2}}=0\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] done clear
  View Solution play_arrow

• question_answer28)
  If A and B are two fixed points and P is a variable point such that \[PA+PB=4\], then the locus of P is a/an [IIT 1989; MNR 1991; UPSEAT 2000]
  A)
  Parabola done clear
  B)
  Ellipse done clear
  C)
  Hyperbola done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer29)
  If A and B are two points in a plane, so that  \[PA-PB\] = constant, then the locus of P is   [MNR 1991, 95]
  A)
  Hyperbola done clear
  B)
  Circle done clear
  C)
  Parabola done clear
  D)
  Ellipse done clear
  View Solution play_arrow

• question_answer30)
  If A and B are two fixed points in a plane and P is another variable point such that \[P{{A}^{2}}+P{{B}^{2}}=\]constant, then the locus of the point P is     [MNR 1991]
  A)
  Hyperbola done clear
  B)
  Circle done clear
  C)
  Parabola done clear
  D)
  Ellipse done clear
  View Solution play_arrow

• question_answer31)
  The locus of P such that area of \[\Delta PAB=12sq.\] units, where \[A(2,3)\] and \[B(-4,5)\] is  [EAMCET 1989]
  A)
  \[(x+3y-1)(x+3y-23)=0\] done clear
  B)
  \[(x+3y+1)(x+3y-23)=0\] done clear
  C)
  \[(3x+y-1)(3x+y-23)=0\] done clear
  D)
  \[(3x+y+1)(3x+y+23)=0\] done clear
  View Solution play_arrow

• question_answer32)
  The position of a moving point in the XY-plane at time t is given by \[\left( (u\cos \alpha )t,(u\sin \alpha )t-\frac{1}{2}g{{t}^{2}} \right),\] where \[u,\,\alpha ,\,g\]are constants. The locus of the moving point is
  A)
  A circle done clear
  B)
  A parabola done clear
  C)
  An ellipse done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer33)
  If \[A(\cos \alpha ,\sin \alpha ),\ B(\sin \alpha ,-\cos \alpha ),\,C(1,\text{ }2)\]are the vertices of a \[\Delta ABC\], then as \[\alpha \]varies, the locus of its centroid is   
  A)
  \[{{x}^{2}}+{{y}^{2}}-2x-4y+1=0\] done clear
  B)
  \[3({{x}^{2}}+{{y}^{2}})-2x-4y+1=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}-2x-4y+3=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer34)
  If the equation of the locus of a point equidistant from the points \[({{a}_{1}},{{b}_{1}})\] and \[({{a}_{2}},{{b}_{2}})\] is \[({{a}_{1}}-{{a}_{2}})x+({{b}_{1}}-{{b}_{2}})y+c=0\], then the value of c is     
  A)
  \[a_{1}^{2}-a_{2}^{2}+b_{1}^{2}-b_{2}^{2}\] done clear
  B)
  \[\sqrt{a_{1}^{2}+b_{1}^{2}-a_{2}^{2}-b_{2}^{2}}\] done clear
  C)
  \[\frac{1}{2}(a_{1}^{2}+a_{2}^{2}+b_{1}^{2}+b_{2}^{2})\] done clear
  D)
  \[\frac{1}{2}(a_{2}^{2}+b_{2}^{2}-a_{1}^{2}-b_{1}^{2})\] done clear
  View Solution play_arrow

• question_answer35)
  If sum of distances of a point from the origin and lines \[x=2\] is 4, then its locus is       [RPET 1997]
  A)
  \[{{x}^{2}}-12y=36\] done clear
  B)
  \[{{y}^{2}}+12x=36\] done clear
  C)
  \[{{y}^{2}}-12x=36\] done clear
  D)
  \[{{x}^{2}}+12y=36\] done clear
  View Solution play_arrow

• question_answer36)
  The locus of a point whose difference of distance from points (3, 0) and (-3,0) is 4, is     [MP PET 2002]
  A)
  \[\frac{{{x}^{2}}}{4}-\frac{{{y}^{2}}}{5}=1\] done clear
  B)
  \[\frac{{{x}^{2}}}{5}-\frac{{{y}^{2}}}{4}=1\] done clear
  C)
  \[\frac{{{x}^{2}}}{2}-\frac{{{y}^{2}}}{3}=1\] done clear
  D)
  \[\frac{{{x}^{2}}}{3}-\frac{{{y}^{2}}}{2}=1\] done clear
  View Solution play_arrow

• question_answer37)
  Locus of centroid of the triangle whose vertices are \[(a\cos t,a\sin t),\ (b\sin t,-b\cos t)\] and (1, 0), where t is a parameter; is   [AIEEE 2003]
  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_answer38)
  If the distance of any point P from the point \[A(a+b,a-b)\] and \[B(a-b,a+b)\]are equal, then the locus of P is   [Karnataka CET 2003]
  A)
  \[x-y=0\] done clear
  B)
  \[ax+by=0\] done clear
  C)
  \[bx-ay=0\] done clear
  D)
  \[x+y=0\] done clear
  View Solution play_arrow

• question_answer39)
  What is the equation of the locus of a point which moves such that 4 times its distance from the x-axis is the square of its distance from the origin    [Karnataka CET 2004]
  A)
  \[{{x}^{2}}+{{y}^{2}}-4y=0\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}-4|y|=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}-4x=0\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}-4|x|=0\] done clear
  View Solution play_arrow

• question_answer40)
  Let P be the point (1, 0) and Q a point of the locus\[{{y}^{2}}=8x\]. The locus of midpoint of PQ is [AIEEE 2005]
  A)
  \[{{x}^{2}}+4y+2=0\] done clear
  B)
  \[{{x}^{2}}-4y+2=0\] done clear
  C)
  \[{{y}^{2}}-4x+2=0\] done clear
  D)
  \[{{y}^{2}}+4x+2=0\] done clear
  View Solution play_arrow