If the tangent at a point P(α,β) on the hyperbola x225−y216=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 Cβ2 Equation of tangent at P(α,β) on the hyperbola x225−y216=1 is αx25−βy16=1 ⇒x=25α+2516⋅βαy…(1) As point P(α,β) lies on the hyperbola so, α225−β216=1…(2)
Tangent line (1) intersects the circle x2+y2=25 Then (25α+2516⋅βαy)2+y2=25 ⇒(1+(2516⋅βα)2)y2+2×25α×25β16αy+(25α)2−25=0 Let the roots of above quadratic equation are y1,y2. y1y2=(25α)2−25(1+(2516⋅βα)2)…(3) y1+y2=−2×25α×25β16α(1+(2516⋅βα)2)…(4) By equation (3) and (4) y1y2y1+y2=(25α)2−25−2×25α×25β16α =(25α)2[1−α225]−2×(25α)2×β16 By equation (2) y1y2y1+y2=−β216−2×β16 ⇒y1y2y1+y2=β2