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Let P and Q be distinct points on the pa...

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?` (a) ( 4 , 2 √ 2 ) (b) ( 9 , 3 √ 2 ) (c) ( 1 4 , 1 √ 2 ) (d) ( 1 , √ 2 )

A

`(4,2sqrt(2))`

B

`(9,3sqrt(2))`

C

`((1)/(4),(1)/(sqrt(2)))`

D

`(1,sqrt(2))`

Text Solution

Verified by Experts

The correct Answer is:
A, D
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`
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