Write the standard equation for the circle that passes through the points:
(0, 0)
(6, 0)
(0, - 8)
You must include the appropriate sign (+ or -) in your answer. Do not use spaces in your answer.
x 2 + y 2 x y = 0
HELP PLEASE I NEED STEP BY STEP EXPLANATION!!
I KNOW THAT THE FORMULA IS =
x^2+y^+dx-ey+f=0
I JUST DON'T GET IT AND NEED SERIOUS EXPLANATION.
Use this standard circle equation:
(x-a)^2 + (y-b)^2 = c^2
Your three points tell you that:
a^2 + b^2 = c^2
(6-a)^2 + b^2 = c^2
a^2 + (b+8)^2 = c^2
The first two of these equations combine to give:
6-a = +or- a . Only + sign works.
a = 3
The first and third equations combine to give
b + 8 = +or- b
Only - sign works.
b = -4
c^2 = a^2 + b^2 = 25
c = +or- 5
So the equation that fits them all is
(x-3)^2 + (y+4)^2 = 25
To find the standard equation for a circle that passes through three given points, we need to follow these steps:
Step 1: Write down the equation of a general circle.
The equation of a circle can be represented as (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center coordinates of the circle, and r represents the radius.
Step 2: Plug in the coordinates of the given points.
We are given three points: (0, 0), (6, 0), and (0, -8).
Using the first point (0, 0):
Substitute x = 0 and y = 0 into the general equation:
(0 - h)^2 + (0 - k)^2 = r^2
Using the second point (6, 0):
Substitute x = 6 and y = 0 into the general equation:
(6 - h)^2 + (0 - k)^2 = r^2
Using the third point (0, -8):
Substitute x = 0 and y = -8 into the general equation:
(0 - h)^2 + (-8 - k)^2 = r^2
Step 3: Simplify the equations.
Expand and simplify the equations obtained in Step 2:
For the first point: h^2 + k^2 = r^2
For the second point: (6 - h)^2 + k^2 = r^2
For the third point: h^2 + (-8 - k)^2 = r^2
Step 4: Solve the system of equations.
Now, we have three equations with three variables (h, k, and r). By solving this system, we can find the specific values of h, k, and r.
From the first equation, we can express r^2 as h^2 + k^2.
Substituting r^2 into the second equation:
(6 - h)^2 + k^2 = h^2 + k^2
Expanding and simplifying:
36 - 12h + h^2 + k^2 = h^2 + k^2
36 - 12h = 0
12h = 36
h = 3
Substituting h = 3 into the third equation:
3^2 + (-8 - k)^2 = 3^2 + k^2
9 + 64 + 16k + k^2 = 9 + k^2
73 + 16k = 9
16k = -64
k = -4
Step 5: Substitute the values of h and k into any of the original equations to find r.
Using the first equation:
(3 - 0)^2 + (-4 - 0)^2 = r^2
9 + 16 = r^2
25 = r^2
r = 5 (taking the positive square root since r represents the radius of a circle)
Step 6: Write the final equation.
Now that we have the values for h, k, and r, we can substitute them into the general equation to obtain the final equation for the circle:
(x - 3)^2 + (y + 4)^2 = 5^2
Therefore, the final standard equation for the circle that passes through the points (0, 0), (6, 0), and (0, -8) is:
(x - 3)^2 + (y + 4)^2 = 25