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Find circle equation from three points on the circle

Question:

Three points A (-2, 0), B (8, 0), and C (0, 5) lie on a circle. Find the equation of the circle.

how to find circle equation from three points on the circle?

Solution:

The general form of the circle equation is: x2 + y2 + DX + EY + F = 0. Because points A, B and C lie on the circle, then points A, B and C satisfy the circle equation.

Substitute point A (-2, 0) into the circle equation, we get 4 - 2D + F = 0, name this as equation1

Substitute point B (8, 0) into the circle equation, we get 64 + 8D + F = 0, name this as equation2

Substitute point C (0, 5) into the circle equation, we get 25 + 5E + F = 0, name this as equation3

Now we want to remove the variable D, so we use equation1 × 4, we get 16 - 8D + 4F = 0, name this as equation4

Use equation4 + equation2 to remove variable D, we get 80 + 5F = 0 -> F = -80/5 = -16

Substitute the value of F into the equation3 to get the value of E. 25 + 5E - 16 = 0 -> 5E = 16 - 25 = -9 -> E = -9/5

Substitute the value of F into the equation1 to get the value of D. 4 - 2D - 16 = 0 -> 2D = 4 - 16 = -12 -> D = -6

Therefore, we get D = -6, E = -9/5, and F = - 16. The circle equation is x2 + y2 - 6x - (9/5)y - 16 = 0

Now we will change the general form of the circle equation into the center and radius form of the circle equation.

x2 + y2 - 6x - (9/5)y - 16 = 0
x2 - 6x + 32 - 32 + y2 - (9/5)y + (9/10)2 - (9/10)2 = 16
(x - 3)2 + (y - 9/10)2 = 16 + 32 + (9/10)2
(x - 3)2 + (y - 9/10)2 = 16 + 9 + (81/100)
(x - 3)2 + (y - 9/10)2 = 25.81
(x - 3)2 + (y - 9/10)2 = 5.08042

Therefore, the center of the circle is (3, 9/10). The radius of the circle is r = 5.0804. Watch the video for more details.