A Tangent That Measures Distance

Problem

Consider the line $\ell: y = x + t$ and the circle $C: x^2 + (y-2)^2 = 8$ .

Suppose there is a fixed point $M$ such that for every point $P$ on $\ell$ , the tangent length from $P$ to the circle equals the distance $|PM|$ — that is, $|PQ| = |PM|$ where $Q$ is the point of tangency.

Find all possible values of $t$ .

Answer

Solution

Difficulty	7/10
Topics	Circles, analytic geometry

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