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let $P$ be the point (1, 0) and $Q$ a point on the locus $y^{2}=8x$ . The locus of mid point of $PQ$ is

Option 1)
$x^{2}-4y+2=0$

Option 2)
$x^{2}+4y+2=0$

Option 3)
$y^{2}+4x+2=0$

Option 4)
$y^{2}-4x+2=0$

Answers (1)

best_answer

As we learnt in

Parametric coordinates of parabola -

$x= at^{2}$

$y= 2at$

- wherein

For the parabola.

$y^{2}=4ax$

P(1,0) and let the Parametric Coordinate of Q be $(2t^{2}, 4t)$

Mid point PQ is $\left ( \frac{2t^{2}+1}{2} ,\frac{4t}{2}\right )$

Hence, $h=t^{2}+\frac{1}{2}$ and k =2t

$h=\left (\frac{k}{2} \right )^{2}+\frac{1}{2}$

$4h=k^{2}+1$

Replace (h,k) with (x,y)

$y^{2}-4x+1=0$

Option 1)

$x^{2}-4y+2=0$

Incorrect

Option 2)

$x^{2}+4y+2=0$

Incorrect

Option 3)

$y^{2}+4x+2=0$

Incorrect

Option 4)

$y^{2}-4x+2=0$

Correct

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