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Vector Algebra

Question
CBSEENMA12036043
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Let O be the vertex and Q be nay point on the parabola x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1:3 then the locus of P is 

  • x2= y

  • y2 =x

  • y2 =2x

  • x2 = 2y

Solution

D.

x2 = 2y

Any point on the parabola x2 = 8y is (4t, 2t2). Point P divides the line segment joining of O (0,0) and Q (4t,2t2) in the ratio 1:3 Apply the section formula for the internal division.
the equation of the parabola is
x2 = 8y
Let any Q on the parabola (i) is (4t, 2t2).
Let P (h,k) be the point which divides the line segment joining (0,0) and (4t, 2t2) in the ratio 1:3.

straight h space equals space fraction numerator 1 space straight x space 4 straight t plus 3 straight x 0 over denominator 4 end fraction straight h space equals straight t straight k space equals space fraction numerator 1 space straight x space 2 straight t squared straight x 3 space straight x 0 over denominator 4 end fraction space rightwards double arrow space straight k space equals space straight t squared divided by 2 rightwards double arrow space straight k space equals space 1 half straight h squared 2 straight k space equals space straight h squared space rightwards double arrow space 2 straight y space equals space straight x squared

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