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 2` , then which of the following is (are) the coordinates of `P?`

Options

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

Answer

A, D

Explanation

1,4
`OPbotOQ`
`rArrt_(1)t_(2)=-4`
`rArr" Area of "DeltaOPQ=(1)/(2)OP*OQ`

`rArr|(1)/(2)sqrt((t_(1)^(4))/(4)+t_(1)^(2))sqrt((t_(2)^(4))/(4)+t_(2)^(2))|=3sqrt(2)`
`rArr*4*sqrt(((t_(1)^4+4))/(4)((t_(2)^(2)+4))/(4))=3sqrt(2)`
`rArr4*((16+4(t_(1)^(2)+t_(2)^(2))+16))/(16)=9xx2`
`rArr8+t_(1)^(2)+t_(2)^(2)=18`
`rArrt_(1)^(2)+t_(2)^(2)-10=0`
`rArrt_(1)^(4)-10t_(1)^(2)+16=0`
`rArrt_(1)^(2)=2,8`