If a circle passing through the point (-1,0) touches the y-axis at (0,2), then the length …

Question

If a circle passing through the point $(-1,0)$ touches the $y$-axis at $(0,2)$, then the length of the chord of the circle along the $x$-axis is:

Accepted Answer

Since the circle touches the $y$-axis at $(0,2)$, its centre is $(h,2)$ with radius $|h|$:
$(x-h)^2+(y-2)^2=h^2.$
Through $(-1,0)$: $(1+h)^2+4=h^2\Rightarrow 2h+5=0\Rightarrow h=-	frac52$, so $r=	frac52$.

Circle: $\left(x+	frac52ight)^2+(y-2)^2=	frac{25}{4}$. Setting $y=0$: $\left(x+	frac52ight)^2=	frac{9}{4}\Rightarrow x=-1	ext{ or }-4$.

Chord length $=|-1-(-4)|=3.$

Solutions · 1