Q. 36 - Geometry

Question

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

 The circle has center (-2, -4) and passes through point (3, 8).

Step-by-Step Solution

Answer

The equation of the circle is${\left(x+2\right)}^2+{\left(y+4\right)}^2=169$

The graph is:

Step-1 – Given

Given that the circle has center = (-2, -4) and passes through the point (3, 8).

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 (-2, -4) and (3, 8).

$\begin{array}{l}r=\sqrt{{\left(3-\left(-2\right)\right)}^2+{\left(8-\left(-4\right)\right)}^2}\\ r=\sqrt{{\left(3+2\right)}^2+{\left(8+4\right)}^2}\\ r=\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ r=\sqrt{25+144}\\ r=\sqrt{169}\\ r=13\end{array}$

Here, center = (a, b) = (-2, -4) and radius = r = 13.

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

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

${\left(x+2\right)}^2+{\left(y+4\right)}^2={13}^2$

${\left(x+2\right)}^2+{\left(y+4\right)}^2=169$

So, the equation of the circle is ${\left(x+2\right)}^2+{\left(y+4\right)}^2=169$.

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 ${\left(x+2\right)}^2+{\left(y+4\right)}^2=169$ .

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 (-2, -4) and the radius is 13.