Two circles with equal radii are intersecting at the points (0,1) and (0,-1).

Question

Two circles with equal radii are intersecting at the points $(0,1)$ and $(0,-1)$. The tangent at the point $(0,1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

Accepted Answer

By symmetry, centres are $C_1=(d,0)$, $C_2=(-d,0)$, radius $r$ with $r^2=d^2+1$.

Tangent to circle 1 at $(0,1)$ is perpendicular to $C_1$--$(0,1)$, i.e., has direction $(1,d)$: parametrize $(t,1+dt)$. This passes through $C_2=(-d,0)$: $t=-d$, $1-d^2=0\Rightarrow d=1$.

$$\text{Distance}=2d=2.$$

Solutions · 1