How to find the Equation of a Circle if the coordinates of its center and the length of its radius are given.
EXAMPLE 1: What is the equation of a circle whose center is at (-5, -8) and radius is 11 units long?
SOLUTION:
The standard equation of a circle is given by:
(x – h)² + (y – k)² = r² ————————(1)
Where (h, k) is the coordinates of center of the circle and r is the radius of the circle.
In this question, circle center is at (-5, -8)
So,
(h , k) = ( -5, -8)
which implies
h = -5
k = -8
Its radius is 11. So,
r = 11
Put these values in equation (1)
(x – h)² + (y – k)² = r²
{x – (-5) }² + { y – (-8) }² = 11²
{x + 5 }² + { y + 8 }² = 121
This is the standard equation of the circle whose center is at (-5, -8) and radius is 11 units long.
This standard equation can be written as general equation (or simply an equation) of a circle whose center is at (-5, -8) and radius is 11 units long by squaring the terms, then simplifying.
{x + 5 }² + { y + 8 }² = 121
{x² + 25 + 10x} + { y² + 64 + 16y } = 121
x² + y² +10x + 16y + 25 + 64 – 121 = 0
x² + y² +10x + 16y – 32 = 0
This is the general equation (or simply an equation) of a circle whose center is at (-5, -8) and radius is 11 units long.
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