Question

the point (16, 12) lies on a circle that is centered at the origin. create an equation to model the circle.
$(x + \square)^2 + (y + \square)^2 = \square$

Explanation:

Step1: Recall circle equation formula

The standard equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). Since the circle is centered at the origin, \(h = 0\) and \(k = 0\). So the equation becomes \((x - 0)^2 + (y - 0)^2 = r^2\), which simplifies to \(x^2 + y^2 = r^2\).

Step2: Find the radius squared

The point \((16, 12)\) lies on the circle, so we can substitute \(x = 16\) and \(y = 12\) into the equation to find \(r^2\). Substituting these values, we get \(16^2 + 12^2 = r^2\). Calculating \(16^2 = 256\) and \(12^2 = 144\), then adding them together: \(256 + 144 = 400\). So \(r^2 = 400\).

Step3: Write the final equation

Substituting \(h = 0\), \(k = 0\), and \(r^2 = 400\) back into the standard form, we have \((x - 0)^2 + (y - 0)^2 = 400\), which is \((x + 0)^2 + (y + 0)^2 = 400\) (since subtracting 0 is the same as adding 0).

Answer:

\((x + \boldsymbol{0})^2 + (y + \boldsymbol{0})^2 = \boldsymbol{400}\)