Question

Find the center and radius of the circle, and sketch its graph.

${\left(x+1\right)}^2+{\left(y+2\right)}^2=36$

Short answer

The center of the circle is$\left(-1,-2\right)$ and radius is 6 units. The graph of the circle is provided below.

Step-by-step solution

Step 1. Apply the concept of circles.

The standard form of the equation of the circle is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$, where$\left(h,k\right)$ denote the center of the circle andr denote the radius.

Step 2. Interpret the values of center and radius.

Consider the provided equation.

${\left(x+1\right)}^2+{\left(y+2\right)}^2=36$

Rewrite the equation as provided below.

${\left(x-\left(-1\right)\right)}^2+{\left(y-\left(-2\right)\right)}^2={\left(6\right)}^2$

Recall that the standard form of the equation of the circle is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$, where $\left(h,k\right)$ denote the center of the circle and r denote the radius.

Compare, $\left(h,k\right)$and $\left(-1,-2\right)$ also radius r is 6.

Here, $h=-1,k=-2\text{and}r=6$.

Therefore, the center of the circle is $\left(-1,-2\right)$ and radius is 6 units.

Step 3. Graph the circle.

Now, on the Cartesian plane construct a circle with center at$\left(-1,-2\right)$ and radius of 6 units.

The graph obtained is$\left(-1,-2\right)$ provided below,