CircleHardBloom L3
Question
The equation of the smallest circle passing through the points of intersection of the line $x + y = 1$ and the circle $x^{2} + y^{2} = 9$ is:
Options
A.$x^{2} + y^{2} + x + y - 8 = 0$
B.$x^{2} + y^{2} - x - y - 8 = 0$
C.$x^{2} + y^{2} - x + y - 8 = 0$
D.$x^{2} + y^{2} - x - y + 8 = 0$
Solution
[Key Concept: Smallest circle through two circles' intersection: take the family $S_{1} + \lambda\ S_{2} = 0$ and minimise the radius.]
$S \equiv x^{2} + y^{2} - 9 + \lambda(x + y - 1)$
$$\text{~}\text{Centre}\text{~} = \left( \frac{- \lambda}{2},\frac{- \lambda}{2} \right)$$
Centre lies on $x + y = 1$
$$\begin{matrix} \Rightarrow \ & \ - \lambda = 1 \\ & S \equiv x^{2} + y^{2} - x - y - 8 = 0 \end{matrix}$$
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