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Question

Let PQ be a chord of the ellipse x2a2+y2b2=1 which subtends right angle at the centre (0,0). Then its distance from the centre is equal to

A
aba2+b2
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B
aa2+b2
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C
ba2+b2
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D
2aba2+b2
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Solution

The correct option is A aba2+b2
Let xcosα+ysinα=p be the chord PQ.
Then p is the desired distance.
Homogenizing the equation of the ellipse with the help of this equation, we get the combined equation of OP and OQ.

x2a2+y2b2=1
x2a2+y2b2=(xcosα+ysinαp)2
(1a2cos2αp2)x2+(1b2sin2αp2)y22xysinαcosαp2=0
As OPOQ,
coefficient of x2 + coefficient of y2=0 (condition of perpendicular lines)
1a2cos2αp2+1b2sin2αp2=0
1p2=1a2+1b2
p=aba2+b2

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