Question

Question: In Exercises 35-38 find an equation of the circle described and sketch the graph.

The circle has center (0, 6) and passes through point (6, 14).

Short answer

The equation of the circle is$x^2+{\left(y-6\right)}^2=100$ .

The graph is:

Step-by-step solution

Step-1 – Given

Given that the circle has center = (0, 6) and passes through the point (6, 14).

Step-2 – To determine

We have to find the equation of the circle and sketch the graph.

Step-3 – Calculation

We first find the length of the radius by finding the distance between (0, 6) and (6, 14).

$\begin{array}{l}r=\sqrt{{\left(6-0\right)}^2+{\left(14-6\right)}^2}\\ r=\sqrt{{\left(6\right)}^2+{\left(8\right)}^2}\\ r=\sqrt{36+64}\\ r=\sqrt{100}\\ r=10\end{array}$

Here, center = (a, b) = (0, 6) and radius = r = 10.

We plug them in the standard form of the equation of a circle:

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

${\left(x-0\right)}^2+{\left(y-6\right)}^2={10}^2$

$x^2+{\left(y-6\right)}^2=100$

So, the equation of the circle is $x^2+{\left(y-6\right)}^2=100$.

Step-4 – Graph

We willsketch the graph using a graphing utility.

Step 1: Press WINDOW button in order to access the Window editor.

Step 2: Press$\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $x^2+{\left(y-6\right)}^2=100$.

Step 4: Press GRAPH button to graph the function and then adjust the window.

The obtained graph is:

From the graph, we see that the center is (0, 6) and the radius is 10.