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If the tangent at point P(h, k) on the h...

If the tangent at point P(h, k) on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` cuts the circle `x^(2)+y^(2)=a^(2)` at points `Q(x_(1),y_(1))` and `R(x_(2),y_(2))`, then the value of `(1)/(y_(1))+(1)/(y_(2))` is

Text Solution

Verified by Experts

The correct Answer is:
1
Equation of tangent `(xh)/(a^(2))-(2h)/(b^(2))-1=0,x=(1+(2y)/(b^(2)))(a^(2))/h`
By solving `((1+(2y)/(b^(2)))(a^(2))/h)^(2)+y^(2)-a^(2)=0`
`(1+(4y)/(b^(2))+(4y^(2))/(b^(4)))(a^(4))/(h^(2))+y^(2)-a^(2)=0`
`y^(2) ((4a^(4))/(b^(4)h^(2))+1)+4/(b^(2)).(a^(4))/(h^(2)).y+(a^(4))/(h^(2))-a^(2)=0`
`implies (y_(1)+y_(2))/(y_(1)y_(2))=1`
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