Find the equation of the circle that passes through the points $ A(1, 2) $, $ B(4, 6) $, and $ C(-3, -5) $

To find the equation of the circle passing through the points $ A(1, 2) $, $ B(4, 6) $, and $ C(-3, -5) $, we can use the determinant method. The equation of a circle is given by:

$ x^2 + y^2 + Dx + Ey + F = 0 $

Substituting the given points into the equation, we get three equations:

1. $ 1^2 + 2^2 + D(1) + E(2) + F = 0 $

2. $ 4^2 + 6^2 + D(4) + E(6) + F = 0 $

3. $ (-3)^2 + (-5)^2 + D(-3) + E(-5) + F = 0 $

Which simplifies to:

1. $ 1 + 4 + D + 2E + F = 0 $

2. $ 16 + 36 + 4D + 6E + F = 0 $

3. $ 9 + 25 – 3D – 5E + F = 0 $

Rearranging these equations, we get:

1. $ D + 2E + F = -5 $

2. $ 4D + 6E + F = -52 $

3. $ -3D – 5E + F = -34 $

Solving this system of linear equations will give us the values of D, E, and F.

For the circle passing through points $ A(1, 2) $, $ B(4, 6) $, and $ C(-3, -5) $, we start with the general form of the circle

To derive the equation of the circle through $ A(1, 2) $, $ B(4, 6) $, and $ C(-3, -5) $:

$ x^2 + y^2 + Dx + Ey + F = 0 $

Substitute $ (1, 2) $:

$ D + 2E + F = -5 $

Substitute $ (4, 6) $:

$ 4D + 6E + F = -52 $

Substitute $ (-3, -5) $:

$ -3D – 5E +

Start Using PopAi Today