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Question

From any point on the hyperbola x2a2y2b2=1, tangents are drawn to the hyperbola x2a2y2b2=2 . Then, area cut-off by the chord of contact on the asymptotes is equal to

A
a/2 sq unit
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B
ab sq unit
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C
2ab sq unit
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D
4ab sq unit
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Solution

The correct option is D 4ab sq unit
Let P(x1,y1) be a point on the hyperbola x2a2+y2b2=1
The chord of contact of tangents from P to the hyperbola is given by xx1a2+yy1b2=1 …… (i)
The equation of the asymptotes are xayb=0
and xa+yb=0
The points of intersection of Equation (i) with the two asymptotes are given by
x1=2ax1a+y1b,y1=2ax1a+y1b
x2=2ax1a+y1b,y2=2ax1a+y1b
Area of the triangle = 12(x1x2x2y1)
=12∣ ∣ ∣4ab×2x21a2y21b2∣ ∣ ∣

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