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Applications of Cross Product

If the circle...

Question

If the circle $x^{2} + y^{2} + 4 x + 22 y + c = 0$ bisects the circumference of the circle $x^{2} + y^{2} - 2 x + 8 y - d = 0$ then $c + d$ is equal to

50

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64

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25

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Solution

The correct option is A $50$
The common chord must be a diameter of the second circle.
The equation to the common chord is
$S_{1} - S_{2} = 0,$ i.e., $6 x + 14 y + (c + d) = 0.$
The centre of the second circle $(1, - 4)$ lies on it,
$∴ 6 (1) + 14 (- 4) + c + d = 0$
$∴ c + d = 50.$

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