Find the equation of the circle whose center (0,13) and the area = 25pi.
Given the center of a circle is the point (0, 13). Also, given that the area of the circle = 25*pi.
We will write the equation of the circle.
(x-a)^2 + (y-b)^2 = r^2 where (a,b) is the center and r is the radius.
==> (x-0)^2 + (y-13)^2 = r^2
==> x^2 + ( y-13)^2 = r^2.
Now we need to determine the radius.
Given the area of the circle = 25pi.
==> r^2 * pi = 25*pi
==> r^2 = 25
==> r= 5
Then the radius of the circle = 5
==> x^2 + ( y-13)^2 = 5^2
==> x^2 + ( y-13)^2 = 25
==> x^2 + y^2 - 26y + 144 = 0
We'll write the equation of the circle in standard form:
(x-h)^2 + (y-k)^2 = r^2, where 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 determine the radius of the circle, using the formula for area of the circle:
A = pi*r^2
25*pi = pi*r^2
We'll divide by pi:
r^2 = 25
r = 5
We'll accept only the positive value, since it is about a radius of a circle and it cannot be negative.
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
The area of the circle with radius r is pir^2.
Therefore if the actual area is 25pi, then pir^2 = pi*25 => r^2= 25 => r= 5.
So the radius r = 5 and the centre is C(0,13)
The equation of the circle with centre C(h,k) and radius r is (x-h)62+(y-k)^2 = r^2.
Therefore the equation of the circle with C(0,13) and r = 5 is
(x-0)^2+(y-13)^2 = 5^2.
x^2+y^2-26y+169 = 25.
x^2+y^2 -26y - 144 = 0 is the equation if the circle.