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Parabola question

2011 · Shift 2 · Q26
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Parabola question

2011 · Shift 2 · Q26

JEE AdvancedMathematicsParabolaMCQ+3 / −0.75
Let (x,y)(x, y)(x,y) be any point on the parabola y2=4x{y^2} = 4xy2=4x. Let PPP be the point that divides the line segment from (0,0)(0, 0)(0,0) to (x,y)(x, y)(x,y) in the ratio 1:31 : 31:3. Then the locus of PPP is
  1. A
    x2=y{x^2} = yx2=y
  2. B
    y2=2x{y^2} = 2xy2=2x
  3. C
    y2=x{y^2} = xy2=x
  4. D
    x2=2y{x^2} = 2yx2=2y
View written solutionFree

Correct answer: C

  1. Let the point on the parabola y2=4xy^2=4xy2=4x be Q(x,y)Q(x,y)Q(x,y).

  2. The point PPP divides the line segment from (0,0)(0,0)(0,0) to (x,y)(x,y)(x,y) in the ratio 1:31:31:3.

    Since PPP divides the segment joining (0,0)(0,0)(0,0) and (x,y)(x,y)(x,y) internally in the ratio 1:31:31:3, its coordinates are P(1⋅x+3⋅01+3,1⋅y+3⋅01+3)=(x4,y4).P\left(\frac{1\cdot x+3\cdot 0}{1+3},\frac{1\cdot y+3\cdot 0}{1+3}\right)=\left(\frac{x}{4},\frac{y}{4}\right).P(1+31⋅x+3⋅0​,1+31⋅y+3⋅0​)=(4x​,4y​).

  3. Let the coordinates of PPP be (X,Y)(X,Y)(X,Y). Then X=x4,Y=y4.X=\frac{x}{4},\qquad Y=\frac{y}{4}.X=4x​,Y=4y​. So, x=4X,y=4Y.x=4X,\qquad y=4Y.x=4X,y=4Y.

  4. Substitute these into the parabola equation y2=4xy^2=4xy2=4x: (4Y)2=4(4X).(4Y)^2 = 4(4X).(4Y)2=4(4X). 16Y2=16X.16Y^2 = 16X. 16Y2=16X. Y2=X.Y^2 = X. Y2=X.

  5. Hence the locus of PPP is Y2=X.Y^2=X.Y2=X. Replacing (X,Y)(X,Y)(X,Y) by the usual (x,y)(x,y)(x,y) notation, the locus is y2=x.y^2=x.y2=x.

  6. Comparing with the options:

    • A: x2=yx^2=yx2=y — incorrect
    • B: y2=2xy^2=2xy2=2x — incorrect
    • C: y2=xy^2=xy2=x — correct
    • D: x2=2yx^2=2yx2=2y — incorrect

Therefore, the correct option is C.

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