Question
A circle in the xy-plane has its center at (-2/3 , -3/4) and radius 5. Which of the following is an equation of the circle?
Answers
GPT-4o mini
The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In your case, the center of the circle is \((-2/3, -3/4)\) and the radius \(r\) is \(5\).
1. Substitute \(h = -\frac{2}{3}\), \(k = -\frac{3}{4}\), and \(r = 5\) into the equation:
\[
(x - (-\frac{2}{3}))^2 + (y - (-\frac{3}{4}))^2 = 5^2
\]
2. This simplifies to:
\[
\left(x + \frac{2}{3}\right)^2 + \left(y + \frac{3}{4}\right)^2 = 25
\]
Thus, the equation of the circle in standard form is:
\[
\left(x + \frac{2}{3}\right)^2 + \left(y + \frac{3}{4}\right)^2 = 25
\]
This is the required equation of the circle.
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In your case, the center of the circle is \((-2/3, -3/4)\) and the radius \(r\) is \(5\).
1. Substitute \(h = -\frac{2}{3}\), \(k = -\frac{3}{4}\), and \(r = 5\) into the equation:
\[
(x - (-\frac{2}{3}))^2 + (y - (-\frac{3}{4}))^2 = 5^2
\]
2. This simplifies to:
\[
\left(x + \frac{2}{3}\right)^2 + \left(y + \frac{3}{4}\right)^2 = 25
\]
Thus, the equation of the circle in standard form is:
\[
\left(x + \frac{2}{3}\right)^2 + \left(y + \frac{3}{4}\right)^2 = 25
\]
This is the required equation of the circle.