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Question

If the tangent at a point P(α,β) on the hyperbola x225y216=1 cuts the circle x2+y2=25 at the point Q(x1,y1) and R(x2,y2), then y1y2y1+y2=

A
2β
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B
β
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C
β2
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D
4β
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Solution

The correct option is C β2
Equation of tangent at P(α,β) on the hyperbola
x225y216=1 is
αx25βy16=1
x=25α+2516βαy (1)
As point P(α,β) lies on the hyperbola so,
α225β216=1 (2)

Tangent line (1) intersects the circle x2+y2=25
Then
(25α+2516βαy)2+y2=25
(1+(2516βα)2)y2+2×25α×25β16αy+(25α)225=0
Let the roots of above quadratic equation are y1,y2.
y1y2=(25α)225(1+(2516βα)2) (3)
y1+y2=2×25α×25β16α(1+(2516βα)2) (4)
By equation (3) and (4)
y1y2y1+y2=(25α)2252×25α×25β16α
=(25α)2[1α225]2×(25α)2×β16
By equation (2)
y1y2y1+y2=β2162×β16
y1y2y1+y2=β2

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