Find the equation of a circle with diameter endpoints of (3,-4) and (10,-12)

Answer 1

An equation for a circle fits the pattern (x - h)² + (y - k)² = r² where (h, k) is the center of the circle and the radius is r.

The center of the circle would be at the midpoint of the diameter. The midpoint is determined by adding the two x values and dividing by 2 and adding the two y values and dividing by 2. So, for the x's: 3 + 10 = 13 and 13/2 = 6.5. For the y's: -4 + -12 = -16 and -16/2 = -8. So the circle center is at the point (6.5, -8).

The radius is half the diameter. To find the length of the diameter, use the distance formula which is:

d = sqrt[(y₂ - y₁)² + (x₂ - x₁)²] = sqrt[(-12- -4)² + (10 - 3)²] = sqrt(64 + 49) = sqrt(113). The radius is the half of that so sqrt(113)/2. The number in the circle equation is the radius squared so that would be [sqrt(113)/2]². [sqrt(113)]² is just 113 and 2² = 4 so the radius squared is 113/4.

The equation is then: (x - 6.5)² + (y + 8)² = 113/4

Answer 2

Use the midpoint formula to find center of the diameter and center of the circle, (h,k), then you use either set of coordinates with that information to solve for the radius

Mipoint is

x₁ + x₂/2 and y₁ + y₂/2

( (10+3)/2, (-12+(-4))/2 ) will give the coordinates of halfway along the diameter and the center of the circle

x = 13/2 =6.5

y = -16/2 = -8

(6.5, -8) are coordinates for half way along the diameter, and the center point of the circle (h, k)

To find r you can plug these values into the Equation for a circle

(x - h)² + (y - k)² = r²

Your circle equation so far is

(x - 6.5)² + (y - (-8))² = r²

Now just plug a set of the coordinates to solve for r

(10 - 6.5)² + (-12 - (-8))² =r²

(3.5)² + (-4)² = r²

12.25 + 16 = r²

28.25 = r²

√28.25 = r

5.32 =r

Finally, your circle equation is

(x - 6.5)² + (y - (-8))² = 28.25

You can check your coordinates for the radius directly with the classic distance formula as well, use either set of the given coordinates with the coordinates of the center

d = sqrt(6.5 - 3)² + ((-8) - (-4))²

d = sqrt(3.5)² + (-4)²

d = sqrt (12.25 + 16) = sqrt(28.25)= 5.3

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