Question

The point (7, 4) lies on a circle centered at the origin. Write the equation of the circle and state the radius.

Answer 1

Given:

The point (7, 4) lies on a circle centered at the origin.

To find:

The equation of the circle and the radius of the circle.

Solution:

Distance formula:

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The point (7, 4) lies on a circle centered at the origin. So, the distance between the points (7,4) and (0,0) is equal to the radius of the circle.

$r=\sqrt{(0-7)^2+(0-4)^2}$

$r=\sqrt{(-7)^2+(-4)^2}$

$r=\sqrt{49+16}$

$r=\sqrt{65}$

The standard form of the circle is

$(x-h)^2+(y-k)^2=r^2$            ...(i)

Where, (h,k) is the center of the circle and r is the radius of the circle.

Putting h=0, k=0 and $r=\sqrt{65}$ in (ii), we get

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

$x^2+y^2=65$

Therefore, the equation of the circle is $x^2+y^2=65$ and its radius is $r=\sqrt{65}$.