Write an equation of the circle described below:
Given a circle with the center (5,1) and a point on the circle (8,-2).
thank you for the help
To find the equation of the circle, we need to use the standard form of a circle equation, which is:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) is the center of the circle and r is the radius.
Given that the center is (5, 1) and a point on the circle is (8, -2), we can use the distance formula to find the radius.
The distance formula is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of the center (5, 1) and the point on the circle (8, -2), we have:
d = sqrt((8 - 5)^2 + (-2 - 1)^2)
d = sqrt(3^2 + (-3)^2)
d = sqrt(9 + 9)
d = sqrt(18)
Therefore, the radius is sqrt(18).
Now, we can plug the values into the standard form equation:
(x - 5)^2 + (y - 1)^2 = (sqrt(18))^2
Simplifying,
(x - 5)^2 + (y - 1)^2 = 18
So, the equation of the circle is:
(x - 5)^2 + (y - 1)^2 = 18
To write the equation of a circle, we use the formula:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the coordinates of the center of the circle, and r is the radius of the circle.
In this case, we are given the center of the circle as (5, 1) and a point on the circle as (8, -2).
Step 1: Determine the radius of the circle
To find the radius of the circle, we need to find the distance between the center of the circle and the given point. The distance formula is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of the center (5, 1) and the given point (8, -2), we have:
Distance = sqrt((8 - 5)^2 + (-2 - 1)^2)
= sqrt(3^2 + (-3)^2)
= sqrt(9 + 9)
= sqrt(18)
Step 2: Substitute the center coordinates and the radius into the equation
Now that we have the radius, we can substitute the values into the equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
Substituting the center coordinates (5, 1) into (h, k) and the radius sqrt(18) into r, we get:
(x - 5)^2 + (y - 1)^2 = 18
Therefore, the equation of the circle is (x - 5)^2 + (y - 1)^2 = 18.
the distance from the center to the point is the radius , find with the distance formula
for a circle centered at (h,k) ... (x - h)^2 + (y - k)^2 = r^2