Mathematics Notes - Analytic Geometry Problems and Solutions

What is the shortest distance between the curve $ x^2 -16x +y^2 +16y +64 =0 $ and the point (26,4)?

Solution:

Identify the curve and locate some important points:

$ x^2 -16x +y^2 +16y = -64 $

$ x^2 -16x +64 +y^2 +16y + 64= -64 +64 +64 $

$ (x-8)^2+(y+8)^2= 8^2 $

This curve is a circle with radius = 8, and center at (8, -8)

$circle$

$ d = \sqrt{(26-8)^2 + (4+8)^2 }$

$ d = 6 \sqrt{13}$

$ d_s = 6 \sqrt{13} - 8 = 13.63 $ units