What is the equation of the circle with center (-6 7) that passes through the point (4 -2)
To find the equation of a circle with the center and a point, we need to make use of the distance formula and complete the square. Here are the steps:
1. Start with the standard form of a circle's equation: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
2. Using the given center (-6, 7), we can substitute these values into the equation. This gives us: (x + 6)^2 + (y - 7)^2 = r^2.
3. Now, we need to find the radius (r). To do this, we use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by: sqrt((x2 - x1)^2 + (y2 - y1)^2).
4. In this case, the distance formula becomes: sqrt((4 - (-6))^2 + (-2 - 7)^2).
5. Simplifying further, we have: sqrt((10)^2 + (9)^2) = sqrt(100 + 81) = sqrt(181).
6. Since the circle passes through the point (4, -2), the distance from the center (-6, 7) to (4, -2) is equal to the radius. Therefore, the radius, r, is sqrt(181).
7. Finally, substituting r = sqrt(181) into the equation gives us the final equation of the circle: (x + 6)^2 + (y - 7)^2 = (sqrt(181))^2.
Simplifying further, the equation of the circle is: (x + 6)^2 + (y - 7)^2 = 181.
To determine the equation of a circle, you need two pieces of information: the coordinates of the center and the radius of the circle.
Since the given circle passes through the point (4, -2), we can use the distance formula to find the radius. The distance between the center (-6, 7) and the point (4, -2) is the radius of the circle.
The distance formula is:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
Plug in the values:
d = √[(4 - (-6))^2 + (-2 - 7)^2]
d = √[10^2 + (-9)^2]
d = √[100 + 81]
d = √181
So, the radius of the circle is √181.
Now that we have the center (-6, 7) and the radius √181, we can write the equation of a circle in standard form:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Plugging in the values:
(x - (-6))^2 + (y - 7)^2 = (√181)^2
(x + 6)^2 + (y - 7)^2 = 181
Therefore, the equation of the circle with center (-6, 7) that passes through the point (4, -2) is (x + 6)^2 + (y - 7)^2 = 181.
X=2
general circle equation ... (x - h)^2 + (y - k)^2 = r^2
... circle centered at (h,k) with radius r
use the distance formula to find the radius ... center to circumference