# What are the conic sections of the following equations x ^2 + y ^2 - 10x -2y +10= 0?

Jan 8, 2016

This is a circle.

#### Explanation:

Complete the squares to find:

$0 = {x}^{2} + {y}^{2} - 10 x - 2 y + 10$

$= \left({x}^{2} - 10 x + 25\right) + \left({y}^{2} - 2 y + 1\right) - 16$

$= {\left(x - 5\right)}^{2} + {\left(y - 1\right)}^{2} - {4}^{2}$

Add ${4}^{2}$ to both ends and transpose to get:

${\left(x - 5\right)}^{2} + {\left(y - 1\right)}^{2} = {4}^{2}$

which is in the form:

${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {r}^{2}$

the equation of a circle, centre $\left(h , k\right) = \left(5 , 1\right)$ and radius $r = 4$

graph{(x^2+y^2-10x-2y+10)((x-5)^2+(y-1)^2-0.01) = 0 [-6.59, 13.41, -3.68, 6.32]}