Questions: Find the equation of the circle with center at the origin that contains the point (-12,-5).

Solution Steps

To find the equation of a circle with a given center and a point on the circle, we need to determine the radius first. The radius is the distance from the center to the given point. Once we have the radius, we can use the standard form of the equation of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

1. Calculate the radius using the distance formula.
2. Substitute the center coordinates and the radius into the standard form of the circle equation.

To find the radius of the circle, we use the distance formula between the center \((0, 0)\) and the point \((-12, -5)\):

\[ r = \sqrt{(-12 - 0)^2 + (-5 - 0)^2} = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13.0 \]

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

Given the center \((0, 0)\) and radius \(13.0\), we substitute these values into the equation:

\[ (x - 0)^2 + (y - 0)^2 = 13.0^2 \]

Simplifying, we get:

\[ x^2 + y^2 = 169.0 \]

Final Answer