Q87 / 160 · AP EAMCET 2006 · Mathematics

Let $O$ be the origin and $A$ be a point on the curve $y^2=4 x$. Then the locus of the mid point of $O A$ is :

  • A
    $x^2=4 y$
  • B
    $x^2=2 y$
  • C
    $y^2=16 x$
  • Correct answer
    D
    $y^2=2 x$

Solution

Since $O$ be the origin and $A$ be the point on the curve $y^2=4 x$. $\therefore$ Co-ordinates of $O$ and $A$ are $(0,0)$ and $\left(a t^2, 2 a t\right)$ respectively. $\therefore$ Co-ordinates of mid point of $O A$ are $\left(\frac{0+a t^2}{2}, \frac{0+2 a t}{2}\right)=\left(\frac{a t^2}{2}, a t\right)$ $\because \quad(a t)^2=2\left(\frac{a t^2}{2}\right)$ Thus the locus of required point is $y^2=2 x$

Hence, the correct answer is option: 'D'.

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