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 is (are) the coordinates of `P?`

Options

A. `(4, 2sqrt2)`
B. `(9, 3sqrt2)`
C. `(1//4, 1//sqrt2)`
D. `(1, sqrt2)`

Answer

A, D

Explanation

Let the parametric coordinates of P and Q be `(t^(2)/2,t)" and "(t_(1)^(2)/2,t_(1))` respectively.
Clearly,
`/_POQ=pi/2`
`rArr" Slope of "Opxx"Slope of "OQ =-1`
`rArr" "t_(1)=-4/1`
So, coordinates of Q are `(8/t^(2), -4t)`
It is given that the area of `DeltaOPQ" is " 3sqrt2` square units.
`:." "1/2(OPxxOQ)=3sqrt2`
`rArr" "1/2xxsqrt(1/4t^(4)+t^(2))xxsqrt((64)/t^(2)+16/t^(2))=3sqrt2`
`rArr" "t^(2)+4=3sqrt2t`
`rArr" "t^(2)-3sqrt2t+4=0`
`rArr" "(t-2sqrt2)(t-sqrt2)=0rArrt=2sqrt2ort=sqrt2`
Hence, the coordinates of P are `(4, 2sqrt2)" or "(1, sqrt2)`.