Question

Find an equation of the circle that satisfies the given conditions.

Center at the origin; passes through$\left(4,7\right)$

Short answer

The equation of the circle is $x^2+y^2=65$.

Step-by-step solution

Step 1. Apply the concept of the distance formula.

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 $d\left(A,B\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

Step 2. Compute the radius of the circle.

Consider the provided conditions that the center of a circle is at $\left(0,0\right)$and passes through the point $\left(4,7\right)$.

Since the circle passes through the point $\left(4,7\right)$, and the center is at $\left(0,0\right)$therefore radius of the circle is the distance between the points $\left(4,7\right)$and $\left(0,0\right)$.

Recall that the 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 $d\left(A,B\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

Apply it.

$$\begin{gathered} d=\sqrt{{\left(4-0\right)}^2+{\left(7-0\right)}^2} \\ =\sqrt{4^2+7^2} \\ =\sqrt{16+49} \\ =\sqrt{65} \end{gathered}$$

Therefore, the radius of the circle is $\sqrt{65}$ units.

Step 3. Apply the concept of circles.

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.

Step 4. Input the values of center and radius.

Consider the conditions that the center of a circle is at $\left(0,0\right)$and radius is $\sqrt{65}$units.

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 $\left(0,0\right)$ also radius r is $\sqrt{65}$.

Here, $h=0,k=0\text{and}r=\sqrt{65}$.

Substitute the values in the standard equation of a circle.

${\left(x-0\right)}^2+{\left(y-0\right)}^2={\left(\sqrt{65}\right)}^2$

Step 5. Calculation step.

Simplify the standard equation of the circle obtained.

$$\begin{gathered} {\left(x-0\right)}^2+{\left(y-0\right)}^2={\left(\sqrt{65}\right)}^2 \\ {x}^2+y^2=65 \end{gathered}$$
Hence, the equation of a circle with center$\left(0,0\right)$ and radius$\sqrt{65}$ is $x^2+y^2=65$.