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# Find the circle equation

Question:

The equation of the circle C is x2 + y2 - 4x + 2y - 15 = 0. Another circle O, its center lies in origin. The circle O is tangent to the circle C. Find the equation of the circle O.

Solution:

Find the center and radius of the circle C.

x2 + y2 - 4x + 2y - 15 = 0
x2 - 4x + 4 - 4 + y2 + 2y + 1 - 1 - 15 = 0
x2 - 4x + 4 + y2 + 2y + 1 = 15 + 4 + 1
(x - 2)2 + (y + 1)2 = 20

Therefore, the center of the circle C is (2, -1) and the radius of the circle C is R = square root of 20.

Find the distance between the centers of two circles. Point C is the center of the circle C. CD is the diameter of the circle C. Point O is the center of the circle O and OD is the radius of the circle O. CD = R, OD = r.

Because the point O is in origin, so, the distance from the point O to the point C is:

|OC| = square root of [22 + (-1)2]
= square root of (4 + 1)
= square root of 5

Because |OC| < R, so the circle O is tangent to the circle C inside the circle C. Their relationship is:

|CD| - |OD| = |OC|
R - r = |OC|

note: R = square root of 20 = square root of 4 × 5 = 2 square root of 5. |OC| = square root of 5. Substitute the values of R and |OC| into the equation R - r = |OC|.

2 square root of 5 - r = square root of 5
r = 2 square root of 5 - square root of 5
= square root of 5

Because the center of the circle O is in origin, so, the circle equation of the circle O is: x2 + y2 = 5 Watch the video for more details.