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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 :
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Verified Answer
The correct answer is:
$y^2=2 x$
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 $y^2=2 x$
$\therefore$ 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 $y^2=2 x$
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