CircleHardBloom L3
Question
From the point $A(0,3)$ on the circle $x^{2} + 4x + (y - 3)^{2} = 0$ a chord $AB$ is drawn and extended to a point $M$ such that $AM = 2AB$. The equation of the locus of $M$ is:
Options
A.$x^{2} + 8x + y^{2} = 0$
B.$x^{2} + 8x + (y - 3)^{2} = 0$
C.$(x - 3)^{2} + 8x + y^{2} = 0$
D.$x^{2} + 8x + 8y^{2} = 0$
Solution
[Key Concept: A chord drawn from a point on the circle and extended: use the power of that point.]
$B \equiv \left( \frac{h}{2},\frac{k + 3}{2} \right)$
Put $B$ to equation of circle
$$\Rightarrow \left( \frac{h}{2} \right)^{2} + 4\left( \frac{h}{2} \right) + \left( \frac{k + 3}{2} - 3 \right)^{2} = 0 $$Required locus is
$$x^{2} + 8x + (y - 3)^{2} = 0 $$
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