circleFind the equation of the circle whose center (0,13) and the area = 25pi.
The area of a circle is pi*r^2. As the area of the circle to be found is 25pi.
pi*r^2 = 25*pi
=> r = 5
The equation of a circle with center (0, 13) and radius 5 is:
(x - 0)^2 + (y - 13)^2 = 5^2
=> x^2 + y^2 + 169 - 26y = 25
=> x^2 + y^2 - 26y + 144 = 0
The equation of the circle is x^2 + y^2 - 26y + 144 = 0
(x-h)^2 + (y-k)^2 = r^2, h and k are the coordinates of the center of the circle.
We'll identify h and k:
h = 0 and k = 13
Now, we'll use the formula of area of the circle to determine the radius of the circle:
A = pi*r^2
25*pi = pi*r^2
We'll divide by pi:
r^2 = 25
r = 5
We'll reject the negative value.
Now,we'll substitute the coordinates of the center of the circle and the value of radius in the equation of the circle:
x^2 + (y - 13)^2 = 25
If we'll expand the square, we'll obtain the general form of the equation:
x^2 + y^2 - 26y + 169 - 25 = 0
x^2 + y^2 - 26y + 144 = 0