Find the standard form of the equation of the circle with the given characteristics.
Endpoints of a diameter: (1, −7) and (7, −5)
well the middle is halfway between
(8/2 , -12/2) = (4,-6) center
so
(x-4)^2 + (y+6)^2 = r^2
try a point like (1,-7)
(-3)^2 + (-7+6)^2 = r^2
9 + 1 = r^2 so r = sqrt 10 =
(x-4)^2 +(y+6)^2 = 10
========================check with other point (7,-5)
(3)^2 + (1)^2 = 9 + 1 = 10 good, checks
To find the standard form of the equation of a circle, we need to know the center and the radius of the circle. We can find the center by finding the midpoint of the given endpoints of the diameter.
The midpoint formula is given by:
Midpoint = [(x1 + x2) / 2, (y1 + y2) / 2]
For the given endpoints (1, -7) and (7, -5), the midpoint is calculated as follows:
Midpoint = [(1 + 7) / 2, (-7 + -5) / 2]
= [8 / 2, -12 / 2]
= [4, -6]
So, the center of the circle is (4, -6).
Now, to find the radius, we can use the distance formula. The diameter of the circle is the distance between the two endpoints, and the radius is half the length of the diameter.
The distance formula is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the given endpoints (1, -7) and (7, -5), the length of the diameter is calculated as follows:
Diameter = sqrt((7 - 1)^2 + (-5 - (-7))^2)
= sqrt(6^2 + 2^2)
= sqrt(36 + 4)
= sqrt(40)
= 2 * sqrt(10)
Therefore, the radius is half the length of the diameter, so the radius is:
Radius = (2 * sqrt(10)) / 2
= sqrt(10)
Now that we have the center and the radius, we can write the equation of the circle in standard form.
The standard form of the equation of a circle is given by:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Substituting the values, we have:
(x - 4)^2 + (y + 6)^2 = (sqrt(10))^2
(x - 4)^2 + (y + 6)^2 = 10
Therefore, the standard form of the equation of the circle with the given characteristics is:
(x - 4)^2 + (y + 6)^2 = 10