The coordinates of the centre of a circle is (-2, 3). If its area is \(25\pi cm^{2}\), find its equation.
Answer Details
We know that the equation of a circle with center coordinates \((a,b)\) and radius \(r\) is given by \((x-a)^2 + (y-b)^2 = r^2\).
From the problem statement, we are given that the center of the circle is at \((-2,3)\), so we can substitute \(a=-2\) and \(b=3\) in the equation of the circle as \((x+2)^2 + (y-3)^2 = r^2\).
We are also given that the area of the circle is \(25\pi cm^2\). We know that the area of a circle is given by the formula \(A = \pi r^2\), where \(r\) is the radius of the circle. We can solve for the radius by substituting the area value, so we have
\begin{align*}
A &= \pi r^2\\
25\pi &= \pi r^2\\
25 &= r^2\\
r &= 5
\end{align*}
Now we have the coordinates of the center and the radius, so we can substitute them in the equation of the circle to get
$$(x+2)^2 + (y-3)^2 = 25$$
Expanding the square terms, we have
$$x^2+4x+4+y^2-6y+9=25$$
Simplifying, we get
$$x^2+y^2+4x-6y-12=0$$
Therefore, the equation of the circle is \boxed{x^2+y^2+4x-6y-12=0}.