To find the equation of a circle given its center and a point on the circle, we can use the formula:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) represents the center of the circle, and r is the radius. In this case, the center of the circle is given as (1, 5), and the point (-2, -1) lies on the circle.
To find the radius, we can use the distance formula between the center and the given point:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates (1, 5) and (-2, -1), we can calculate:
r = sqrt((-2 - 1)^2 + (-1 - 5)^2)
= sqrt((-3)^2 + (-6)^2)
= sqrt(9 + 36)
= sqrt(45)
= 3 * sqrt(5)
Now, substituting the values of the center (h, k) = (1, 5) and the radius r = 3 * sqrt(5) into the equation, we get:
(x - 1)^2 + (y - 5)^2 = (3 * sqrt(5))^2
(x - 1)^2 + (y - 5)^2 = 9 * 5
(x - 1)^2 + (y - 5)^2 = 45
Therefore, the equation of the circle in general form with the center at (1, 5) and passing through (-2, -1) is:
Answers & Comments
Answer:
(x - 1)^2 + (y - 5)^2 = 45
Step-by-step explanation:
To find the equation of a circle given its center and a point on the circle, we can use the formula:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) represents the center of the circle, and r is the radius. In this case, the center of the circle is given as (1, 5), and the point (-2, -1) lies on the circle.
To find the radius, we can use the distance formula between the center and the given point:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates (1, 5) and (-2, -1), we can calculate:
r = sqrt((-2 - 1)^2 + (-1 - 5)^2)
= sqrt((-3)^2 + (-6)^2)
= sqrt(9 + 36)
= sqrt(45)
= 3 * sqrt(5)
Now, substituting the values of the center (h, k) = (1, 5) and the radius r = 3 * sqrt(5) into the equation, we get:
(x - 1)^2 + (y - 5)^2 = (3 * sqrt(5))^2
(x - 1)^2 + (y - 5)^2 = 9 * 5
(x - 1)^2 + (y - 5)^2 = 45
Therefore, the equation of the circle in general form with the center at (1, 5) and passing through (-2, -1) is:
(x - 1)^2 + (y - 5)^2 = 45