# How do you determine circle, parabola, ellipse, or hyperbola from equation x^2 + y^2 - 16x + 18y - 11 = 0?

Nov 30, 2015

The equation is of a circle of radius $\sqrt{156}$ centered at $\left(8 , - 9\right)$

#### Explanation:

Step 1: Group $x$'s and $y$'s

${x}^{2} - 16 x + {y}^{2} + 18 y = 11$

Step 2: Complete the square for both $x$ and $y$

${x}^{2} - 16 x + 64 + {y}^{2} + 18 y + 81 = 11 + 64 + 81$

$\implies {\left(x - 8\right)}^{2} + {\left(y + 9\right)}^{2} = 156$

Step 3: Compare to the standard forms of conic sections

Note that the above equation matches the formula for a circle with $h = 8$, $k = - 9$, and $r = \sqrt{156}$

Thus the equation is of a circle of radius $\sqrt{156}$ centered at $\left(8 , - 9\right)$