Question

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

${\left(x+3\right)}^2+{\left(y-4\right)}^2=25$

Short answer

The center of the circle is$\left(-3,4\right)$ and radius is 5 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 and r denote the radius.

Step 2. Interpret the values of center and radius.

Consider the provided equation.

${\left(x+3\right)}^2+{\left(y-4\right)}^2=25$

Rewrite the equation as provided below.

${\left(x-\left(-3\right)\right)}^2+{\left(y-4\right)}^2={\left(5\right)}^2$

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 $\left(-3,4\right)$ also radius r is 5.

Here, $h=-3,k=4\text{and}r=5$.

Therefore, the center of the circle is $\left(-3,4\right)$and radius is 5 units.

Step 3. Graph the circle.

Now, on the Cartesian plane construct a circle with a center at $\left(-3,4\right)$and radius 5 units.