A pair of Circles

Geometry Level 3

Consider a pair of circles:

$C_{1}: (x-a)^2+(y-b)^2=r^2$

$C_{2}: (x-c)^2+(y-d)^2=q^2$

Given these two circles meet each other a single point P: $(\alpha_{1},\beta_{1})$

Does the point P satisfy the line $L_{1}: (y-b)(c-a) = (d-b)(x-a)$ ?

Bonus: Prove or disprove it algebraically in the comments below.

Yes No

1 solution

Sid Patak
Dec 4, 2019

$L_{1}: (y-b)(c-a) = (d-b)(x-a)$

Note that this line is the line passing through the centers of the two circles. (Recall point slope formula.)

Statement: If two circles meet at a unique point then the line joining their centers passes through the point of intersection.

Bonus: Prove the statement algebraically using the data in the problem description.