Chapter 9.2, Problem 49WE

To determine

To find: The equation of the circle.

Answer to Problem 49WE

The equation of the circle is $x^2+{\left(y-4\sqrt{2}\right)}^2=16$ .

Explanation of Solution

Given information: The given circle has radius of 4 and it is tangent to the $y=\left|x\right|$ .

Formula used: ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$

Calculation: Since $y=\left|x\right|$ can be also written as $y=\sqrt{x^2}$ .

Now, find the diagonal of the square,

  = the length of a side $\times \sqrt{2}$

  $D=4\sqrt{2}$

Therefore, the center is $\left(h,k\right)=\left(0,4\sqrt{2}\right)$

Now substitute $\left(h,k\right)=\left(0,4\sqrt{2}\right)$ into ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ ,

  $\begin{array}{l}{\left(x-0\right)}^2+{\left(y-4\sqrt{2}\right)}^2=4^2\\ x^2+{\left(y-4\sqrt{2}\right)}^2=16\end{array}$

Therefore, the equation of the circle is $x^2+{\left(y-4\sqrt{2}\right)}^2=16$ .