HOW TO FIND EQUATION OF CIRCLE WITH CENTER AND RADIUS

Problem 1 :

Find the equation of a circle with center (12, -3) and a radius of 3.

Solution :

Here (h, k) is (12, -3).

(x - h)² + (y - k)² = r²

By applying the values of h and k in the equation above, we get

(x - 12)² + (y - (-3))² = 3²

(x - 12)² + (y + 3)² = 9

Expanding using algebraic identities

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

x² - 2x(12) + 12² + y² + 2y(3) + 3² = 9

x² - 24x + 144 + y² + 6y + 9 = 9

x² + y² - 24x + 6y + 144 = 0

Example 2 :

Find the equation of a circle with center (-7, 11) and a radius of 9.

Solution :

Here (h, k) is (-7, 11).

(x - h)² + (y - k)² = r²

By applying the values of h and k in the equation above, we get

(x - (-7))² + (y - 11)² = 9²

(x + 7)² + (y - 11)² = 9²

x² + 2x(7) + 7² + y² - 2y(11) + 11² = 81

x² + 14x + y² - 22y + 49 + 121 = 81

x² + 14x + y² - 22y + 89 = 0

Problem 3 :

Given the center and radius, write the equation.

C (5, 2)  and r = 7

Solution :

Here (h, k) is (5, 2).

(x - h)² + (y - k)² = r²

By applying the values of h and k in the equation above, we get

(x - 5)² + (y - 2)² = 7²

x² - 2x(5) + 5² + y² - 2y(2) + 2² = 7²

x² - 10x + 25 + y² - 4y + 4 = 49

x²+ y²- 10x - 4y + 29 - 49 = 0

x²+ y²- 10x - 4y - 20 = 0

Problem 4 :

Given the center and radius, write the equation.

C (-3, 4) and r = 2

Solution :

Here (h, k) is (-3, 4).

(x - h)² + (y - k)² = r²

By applying the values of h and k in the equation above, we get

(x - (-3))² + (y - 4)² = 2²

(x + 3)² + (y - 4)² = 2²

x² + 2x(3) + 3² + y² + 2y(4) + 4² = 4

x² + 6x + y² + 8y + 9 + 16 - 4 = 0

x² + 6x + y² + 8y + 21 = 0