A Circumcenter on a Line

★ 7/10Circles conic sections analytic geometry Symmetry

Problem

A moving point $Q$ in the coordinate plane satisfies $|QE| - |QF| = 2\sqrt{2}$ , where $E(-\sqrt{3}, 0)$ and $F(\sqrt{3}, 0)$ .

1. Find the equation of the locus $C$ of $Q$ . 2. A line of slope $-1$ meets the curve $C$ at points $A$ and $B$ ; let $P(2, 1)$ .

Find the sum of the slopes of lines $AP$ and $BP$ .
Does the circumcenter $M$ of $\triangle PAB$ lie on a fixed line? Explain.

Answer

Solution

Difficulty	7/10
Topics	Circles, conic sections, analytic geometry, Symmetry

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments

No comments yet.