A circle is the set of all points that are the same distance, r, from a fixed point called center.
If (x, y) is any point on the circle whose center is at (a, b) then the radius r of the circle is:
According to Pythagorean Theorem (Square of hypotenuse is sum of square of other two sides of the right triangle):
(x – a)2 + (y – b)2 = r2
The above equation is called the Standard Form of equation of a circle. This form is also referred as center-radius form of equation of a circle.
The above equation can be simplified to get the General Form of equation of a circle.
(x2 – 2ax + a2) + (y2 – 2by + b2) = r2
x2 + y2 – 2ax – 2by + a2 + b2 = r2
which can also be written as:
x2 + y2 + Ax + by + C = 0
where, A = -2a, B = -2b and C = a2 + b2 – r2
General form of equation of circle can be converted to standard form by using completing the square method.
Example 1:
What is the equation of the circle with radius 5 and center at (-2, 3)?
Here, a = -2, b = 3 and r = 5
Using standard form of equation:
(x – (-2))2 + (y – 3)2 = 52
(x + 2)2 + (y – 3)2 = 25
This can also be written in general form as
x2 + y2 – 2(-2)x – 2(3)y + (-2)2 + 32 = 52
x2 + y2 + 4x – 6y + 4 + 9 = 25
x2 + y2 + 4x – 6y + 13 = 25
x2 + y2 + 4x – 6y – 12 = 0
Example 2:
Convert x2 + y2 – 4x – 2y – 6 = 0 to standard form.
(x2 – 4x) + (y2 – 2y) = 6
(x2 – 4x + 4) + (y2 – 2y + 1) = 6 + 4 + 1
(x – 2)2 + (y – 1)2 = 11
(2, 1) is the center of the circle whose radius is √11.
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