Question

Let $P$ and $Q$ be distinct points on the parabola ${y^2} = 2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\Delta OPQ$ is$3\sqrt 2$, then which of the following are the coordinates of $P$?
A. $\left( {4,2\sqrt 2 } \right)$
B. $\left( {9,3\sqrt 2 } \right)$
C. $\left( {\dfrac{1}{4},\dfrac{1}{{\sqrt 2 }}} \right)$
D.$\left( {4,2\sqrt 2 } \right)or\left( {1,\sqrt 2 } \right)$

Answer 1

Hint: Start by taking a point in the first quadrant and since we have been given that the circle is drawn using PQ as diameter and it is also passing through the vertex therefore take O (0,0), now using the information that PO and OQ are perpendicular, we can use the rule of multiplication of slopes will be equal to -1 and then obtain the relation, the next step is to calculate the area of using matrix operations to get the answer.

Complete step-by-step answer:

Parabola given to us is: ${y^2} = 2x$,
Since it is given that is a point on the first quadrant we’re going to take the point$P\left( {\dfrac{{{t_1}^2}}{2},{t_1}} \right),Q\left( {\dfrac{{{t_2}^2}}{2},{t_2}} \right)$.
The next information given to us is a circle is drawn with $PQ$ as diameter and it passes through the vertex O therefore$O\left( {0,0} \right)$.
$\angle POQ = {90^ \circ }$in .
Since, $PO$and $OQ$are perpendicular.
Therefore,
(Slope of $OP$)$\times$(Slope of $OQ$)$= - 1$
$\Rightarrow \dfrac{2}{{{t_1}}} \times \dfrac{{ - 2}}{{{t_2}}} = - 1 \Rightarrow {t_1}{t_2} = 4.....\left( 1 \right)$

Area of \Delta OPQ = \dfrac{1}{2}\left( {\begin{array}{*{20}{c}} {\dfrac{{{t_1}^2}}{2}}&{{t_1}}&1 \\\ {{a_{21}}}&{ - {t_2}}&1 \\\ 0&0&1 \end{array}} \right) = 3\sqrt 2
${t_1} + {t_2} = 3\sqrt 2 ....\left( 2 \right)$
From (1) and (2), we get,
$\left( {{t_1},{t_2}} \right)$ either $\left( {2\sqrt 2 ,\sqrt 2 } \right)$ or $\left( {\sqrt 2 ,2\sqrt 2 } \right)$
Therefore, the coordinates of are $\left( {4,2\sqrt 2 } \right)or\left( {1,\sqrt 2 } \right)$
Note: To obtain the coordinates of P, we started by forming equations using the conditions given in the questions and then simplified it to get the answer.