Question

Find an equation of the circle with the following characteristics
and sketch its graph.
center on y = 8-x, tangent to both axes

Answer 1

The equation of a circle is used to calculate the radius and the center of the circle.

The equation of the circle is: $\mathbf{(x + 4)\² + (y - 4)\² = 4\²}$

The given parameters are:

$\mathbf{y = 8 - x}$ --- center

Tangent to both axes

The equation of a circle is:

$\mathbf{(x - h)\² + (y - k)\² = r\²}$

where (h, k) is the center and

r is the radius.

If a circle is tangent to both axes, then

$\mathbf{|h| = |k| = r}$

Where h and k have different signs

i.e.

$\mathbf{h = -k}$

This means that:

$\mathbf{y = 8 - x}$

$\mathbf{k = 8 -k}$

$\mathbf{k +k= 8}$

$\mathbf{2k= 8}$

Divide both sides by 2

$\mathbf{k= 4}$

Recall that:

$\mathbf{h = -k}$

This means:

$\mathbf{h = -4}$

Recall that:

$\mathbf{|h| = |k| = r}$

This means that:

$\mathbf{r = 4}$

Recall that:

$\mathbf{(x - h)\² + (y - k)\² = r\²}$

Substitute values for h, k and r

$\mathbf{(x + 4)\² + (y - 4)\² = 4\²}$

Hence, the equation of the circle is:

$\mathbf{(x + 4)\² + (y - 4)\² = 4\²}$

See attachment for the graph of the circle

Read more about equations of circle at:

https://brainly.com/question/13392797