Let P and Q be distinct points on the parabola y^2=2 x such that … - Vidyadip

MCQ

Let $P$ and $Q$ be distinct points on the parabola $y^2=2 x$ such that a circle with $P Q$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\Delta O P Q$ is $3 \sqrt{2}$, then which of the following is (are) the coordinates of $P$ ?

$(A)$ $(4,2 \sqrt{2})$ $(B)$ $(9,3 \sqrt{2})$ $(C)$ $\left(\frac{1}{4}, \frac{1}{\sqrt{2}}\right)$ $(D)$ $(1, \sqrt{2})$

A
$(B,C)$
B
$(B,D)$
C
$(A,C)$
✓
$(A,D)$

✓

Answer

Correct option: D.

$(A,D)$

d
$P\left( at ^2, 2 at \right)$

$Q \left(\frac{16 a }{ t ^2},-\frac{8 a }{ t }\right)$

$\triangle OPQ =\frac{1}{2} OP \cdot OQ$

$\Rightarrow \frac{1}{2}\left| at \sqrt{ t ^2+4} \cdot \frac{ a (-4)}{ t } \sqrt{\frac{16}{ t ^2}+4}\right|=3 \sqrt{2}$

$t ^2-3 \sqrt{2} t +4=0 $

$t =\sqrt{2}, 2 \sqrt{2}$

$\text { Hence, } P ( at , 2 at )= P \left(\frac{ t ^2}{2}, t \right)$

$t =\sqrt{2} \Rightarrow P (1, \sqrt{2}) $

$t =2 \sqrt{2} \Rightarrow P (4,2 \sqrt{2})$

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