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.