Q. 35 - Geometry

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).

Step-by-Step Solution

Answer

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

The graph is:

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 will sketch 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.