Let the equation of the sphere through the circle passing through $x^{2} + y^{2} + z^{2} = 9, 2 x + 3 y + 4 z = 5$ and the point $(1, 2, 3)$ be $k (x^{2} + y^{2} + z^{2}) - m x - n y - r z - p = 0.$ Find $p - k - m - n - r - 1$ ?

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Solution

Sphere through the given circle is given by
$S + λ P = 0.$
or $(x^{2} + y^{2} + z^{2} - 9) + λ (2 x + 3 y + 4 z - 5) = 0.$
It passes through $(1, 2, 3); ∴ λ = - \frac{1}{3} .$
$∴$ Required sphere is
$3 (x^{2} + y^{2} + z^{2}) - 2 x - 3 y - 4 z - 22 = 0$
$\Rightarrow p - k - m - n - r - 1 = 9$

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