Let the equation of the circle which passes through the focus of the parabola $x^{2} = 4 y$ and touches it at the point $(6, 9)$ be $x^{2} + y^{2} + m x - n y + 27 = 0$ . Find $n - m - 1$ ?

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Solution

Focus of the given parabola is $(0, 1)$
Given circle $x^{2} + y^{2} + m x - n y + 27 = 0$ . passes through $(0, 1)$
$\Rightarrow 0 + 1 + 0 - n + 27 \Rightarrow n = 28$
Also given circle passes through $(6, 9)$
$\Rightarrow 36 + 81 + 6 m - 28 \times 9 + 27 = 0 \Rightarrow m = 18$
$∴ n - m - 1 = 9$

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