Write the standard form of the equation of the circle that passes through the points at (4,5) (-2, 3), and (-4,-3).
a. (x-5)^2+(y+4)^2=49
b. (x-3)^2+(y+2)^2=50
c. (x+4)^2+(y-2)^2=36
d. (x-2)^2+(y+2)^2=25
I'm having trouble with solving for the variables...after I make the system of equations, I use elimination and wind up with odd numbers (1/9 for E, 307/9 for F).
If somebody who is good at this could show me their steps it would really help me understand what I'm doing wrong.
Thanks so much in advance, I really appreciate the help! MUCH needed!
Since this is multiple choice, the quickest way to determine the correct solution would be to plug in the given coordinates into each of the solutions.
Well, call center at (xc,yc) and radius r
then
(x-xc)^2 + (y-yc)^2 = r^2
distance from center to those points is the same = r
(4-xc)^2 + (5-yc)^2 = r^2
(-2-xc)^2 + (3-yc^2) = r^2
(-4-xc)^2 + (-3-yc^2) = r^2
16-8xc+xc^2+25-10yc+yc^2 = r^2
4+4xc+xc^2+9-6yc+yc^2 = r^2
16+8xc+xc^2+9+6yc+yc^2 = r^2
-8xc+xc^2-10yc+yc^2 = r^2-41
+4xc+xc^2-6yc+yc^2 = r^2-13
+8xc+xc^2+6yc+yc^2 = r^2-25
12 xc +4 yc = 28 changing sign of first and adding to second
4xc +12 yc = -12 changing sign of second and adding to third
12 xc +4 yc = 28
12 xc +36 yc = -36
-32yc = 64
yc = -2
ok, take it from there
of course it has to be b or d, but solve for xc to tell
4 xc + 12(-2) = -12
4 xc = 12
xc = 3
so center at
(3,-2)
and of form
(x-3)^2 +(y+2)^2 = r^2
Thanks Damon!!
This helped a lot...I was totally backwards on some things.
To find the equation of the circle passing through the given points, you can follow these steps:
1. Write the general equation of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) are the coordinates of the center of the circle, and r is the radius.
2. Plug in the coordinates of one of the given points (x1, y1) into the general equation to get an equation involving only h and k.
Example using (4, 5):
(4 - h)^2 + (5 - k)^2 = r^2
3. Repeat the previous step for two other points, which will give you two equations involving h and k.
Example using (-2, 3):
(-2 - h)^2 + (3 - k)^2 = r^2
4. Simplify the equations obtained in the previous step.
(4^2 - 8h + h^2) + (5^2 - 10k + k^2) = r^2 --> equation (1)
(-2^2 + 4h + h^2) + (3^2 - 6k + k^2) = r^2 --> equation (2)
5. Subtract equation (2) from equation (1) to eliminate r^2.
(4^2 - 8h + h^2) + (5^2 - 10k + k^2) - (-2^2 + 4h + h^2) - (3^2 - 6k + k^2) = 0
Simplifying this equation will involve canceling out terms, collecting like terms, and rearranging the equation.
6. Simplify the equation obtained in the previous step to get it in the standard form.
(48 - 18h + 48k) - (7 - 2h - 36k) = 0 --> equation (3)
100k - 20h + 41 = 0
7. Rearrange equation (3) to match the standard form of the equation of a circle, which is (x - a)^2 + (y - b)^2 = r^2.
20h - 100k = 41 --> equation (4)
8. Compare equation (4) with the answer choices given to find the correct equation.
By following these steps, you should be able to arrive at the correct equation of the circle passing through the given points.