The tangents drawn from the origin to the circle $x^{2} + y^{2} - 2 r x - 2 h y + h^{2} = 0$ are perpendicular then sum of all possible values of $\frac{h}{r}$ is

Question

The tangents drawn from the origin to the circle x 2 + y 2 − 2 r x − 2 h y + h 2 = 0 are perpendicular then sum of all possible values of r h ​ is

TestHub · Accepted Answer

Combined equation of the tangents drawn from $(0, 0)$ to the circle is

$(x^{2} + y^{2} - 2 r x - 2 h y + h^{2}) h^{2} = (- r x - h y + h^{2})^{2} here coefficient of$

$x^{2} + co ff ec i e n t o f y^{2} = 0 \Rightarrow (h^{2} - r^{2}) + (h^{2} - r^{2}) = 0$

$\Rightarrow \frac{h}{r} = \pm 1$

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