Question

write the equation in standard form for the circle with center (0, -3) and radius 10.

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\).

Step2: Identify the values of \(h\), \(k\), and \(r\)

Given the center \((0, -3)\), we have \(h = 0\) and \(k = -3\). The radius \(r = 10\).

Step3: Substitute the values into the standard form equation

Substitute \(h = 0\), \(k = -3\), and \(r = 10\) into \((x - h)^2 + (y - k)^2 = r^2\).
For the \(x\)-term: \((x - 0)^2 = x^2\).
For the \(y\)-term: \((y - (-3))^2=(y + 3)^2\).
For the radius term: \(r^2=10^2 = 100\).
So the equation becomes \(x^2+(y + 3)^2=100\).

Answer:

\(x^2+(y + 3)^2 = 100\)