Chapter 1.8, Problem 100E

To determine

To calculate: The equation of a circle whose radius is $5$ and is tangent to both x -axis and y -axis and lies in first quadrant.

Answer to Problem 100E

The equation of the circle is ${\left(x-5\right)}^2+{\left(y-5\right)}^2=25$ .

Explanation of Solution

Given information:

The radius of the circle is $5$ and is tangent to both x -axis and y -axis and lies in first quadrant.

Formula used:

The standard form of the equation of the circle is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ , where $\left(h,k\right)$ denote the center of the circle and r denote the radius.

Distance $d\left(A,B\right)$ between two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $d\left(A,B\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ .

Calculation:

Consider the provided conditions that radius of the circle is $5$ and is tangent to both x -axis and y -axis and lies in first quadrant.

Since the circle is tangent to x -axis so it touches the x -axis at a single point and y -coordinate is 0.

As the radius of circle is 5 units so point on x -axis through which the circle passes is $\left(5,0\right)$ .

Since the circle is tangent to y -axis so it touches the y -axis at a single point and x -coordinate is 0.

As the radius of circle is 5 units so point on y -axis through which the circle passes is $\left(0,5\right)$ .

So, the center of the circle is $\left(5,5\right)$ .

Recall that the standard form of the equation of the circle is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ , where $\left(h,k\right)$ denote the center of the circle and r denote the radius.

Compare, $\left(h,k\right)$ and $\left(5,5\right)$ also radius r is 5.

Here, $h=5,k=5\text{ and }r=5$ .

Substitute the values in standard equation of circle,

  $\begin{array}{c}{\left(x-5\right)}^2+{\left(y-5\right)}^2=5^2\\ {\left(x-5\right)}^2+{\left(y-5\right)}^2=25\end{array}$

Thus, the equation of circle is ${\left(x-5\right)}^2+{\left(y-5\right)}^2=25$ .