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  • Let O be the vertex and Q be any point on the parabola    . If the point P divi

Let O be the vertex and Q be any point on the parabola  x^{2}=8y  . If the point P divides the line segment OQ internally in the ratio 1:3, then the locus of P is

Option: 1

x^{2}=y


Option: 2

y^{2}=x


Option: 3

y^{2}=2x


Option: 4

x^{2}=2y


Answers (1)

best_answer

Equation of parabola is   x^{2}=8y   —-----------(i)

Let any point Q on the parabola (i) is (4t,2t^{2})

Let  P(h,k) be the point which divides the line segment joining  (0,0) and  (4t,2t^{2})  in the ratio 1:3 ,

\therefore h=\frac{1 \times 4 t+3 \times 0}{4} \Rightarrow h=t

And

\begin{aligned} k= & \frac{1 \times 2 t^2+3 \times 0}{4} \Rightarrow k=\frac{t^2}{2} \\ & \Rightarrow k=\frac{1}{2} h^2 \\ \Rightarrow & 2 k=h^2 \end{aligned}

\Rightarrow 2 y=x^2       

which is required locus.

 

 

 

 

 

Posted by

shivangi.bhatnagar

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