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If the tangent at the point (h, k) to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ cuts the auxiliary circle in points whose ordinates are y1 and y2 then prove that $\frac{1}{y1} +\frac{1}{y2} =\frac{2}{k}$

I used following concept

Equation of Auxiliary Circle is $x^2+ y^2=a^2$

Equation of Tangent : $\frac{y-k}{x-h} =m$

Other equation of Tangent $y=mx + \sqrt((am)^2- b^2)%)$

I tried to eliminate 'm' term but it is getting complicated.

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1 Answer 1

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Assuimng $(h,k)$ lies on the given hyperbola, the equation of the tangent is $$\dfrac{xh}{a^2}-\dfrac{yk}{b^2}=1\iff x=\dfrac{a^2(yk+b^2)}{b^2h}$$

Replace the value of $x$ in $$x^2+y^2=a^2$$ to form a quadratic equation in $y$ whose roots are $y_1.y_2$

Now use Vieta's formula

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  • $\begingroup$ I cross checked it matches the answer $\endgroup$ Aug 20, 2017 at 12:35

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