Write the standard form of the equation of the circle that passes through the points at (4,5), (-2,3), (-4,-3).
I have gotten to the point of the system of equations but do not know know how to proceed.
I have: 4D + 5E + F + 41 =0
-2D + 3E + F + 13 =0
-4D - 3E + F + 25 =0
Any help would be great!
I'm afraid I don't recognise the equations in C, D and E you've written down - but I've just sketched out the problem on a piece of graph paper. If it helps you to check your working, the centre of the circle ought to be at (3, -2) and the radius is sqrt(50). That would make the equation (X-3)² + (Y+2)² = 50², which you can check easily by just feeding in your three sets of coordinates, since all three lie on the circumference of the circle. But just writing that down without the working won't be enough: you'll need to use your own methods to get there. Good luck!
To find the standard form of the equation of a circle that passes through three points, you can use the following steps:
1. Start with the general equation for a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r is the radius.
2. Plug in the coordinates of one of the given points into the equation. Let's use the point (4, 5) first:
(4 - h)^2 + (5 - k)^2 = r^2
3. Expand the equation:
16 - 8h + h^2 + 25 - 10k + k^2 = r^2
4. Simplify and rearrange the terms:
h^2 - 8h + k^2 - 10k + 41 - r^2 = 0
5. Repeat steps 2-4 for the other two points. Let's use (-2, 3) next:
(-2 - h)^2 + (3 - k)^2 = r^2
4 + 4h + h^2 + 9 - 6k + k^2 = r^2
h^2 + 4h + k^2 - 6k + 13 - r^2 = 0
6. Finally, use the third point (-4, -3):
(-4 - h)^2 + (-3 - k)^2 = r^2
16 + 8h + h^2 + 9 + 6k + k^2 = r^2
h^2 + 8h + k^2 + 6k + 25 - r^2 = 0
7. Now you have a system of equations with three equations for h, k, and r^2:
h^2 - 8h + k^2 - 10k + 41 - r^2 = 0
h^2 + 4h + k^2 - 6k + 13 - r^2 = 0
h^2 + 8h + k^2 + 6k + 25 - r^2 = 0
To proceed, you can use any method you are comfortable with to solve this system of equations. It can be solved algebraically by elimination or substitution, or you can use matrix methods such as Cramer's rule or Gaussian elimination. Once you find the values of h, k, and r^2, you can substitute them back into the general equation for a circle to obtain the standard form.