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Question

Let P and Q be distinct points on the parabola y2=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 OPQ is 32 sq. units, then which of the following is/are the coordinates of P?

A
(4,22)
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B
(9,32)
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C
(14,12)
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D
(1,2)
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Solution

The correct option is D (1,2)
Given parabola is y2=2x
Let P=(a22,a) and Q=(b22,b)


Slope of OP,
m1=2a
Slope of OQ,
m2=2b

Since PQ is a diameter of the circle, so POQ=90
m1m2=1ab=4 (1)

Area of the triangle OPQ=32
12×OP×OQ=3212×a44+a2×b44+b2=3212×|ab|2×2×a2+4×b2+4=32(a2+4)(b2+4)=72a2+b2=10 (2)

From (1) and (2), we get
a=2,22
Thus, the coordinates of P are (1,2) and (4,22)

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