The coordinates of the endpoint of diameter AB of circle G are A (1,2) and B (-7,-4) find an equation of circle G

Question

There is a line over the AB

Answer 1

The general equation for a circle is:

 

     (x-h)² + (y-k)² = r²

 

where (h,k) are the coordinates of the center of the circle and r is its radius. To find the particular equation for circle G, then, we need to find the center, (h,k), and the radius, r.  First, let's find the coordinates of the center, (h, k),  The segment AB is a diameter, so the center of the circle lies at the midpoint of AB,  To find its coordinates use the Midpoint Formula:

 

(h,k) = ((x₂-x₁)/2, (y₂-y₁)/2)

 

where (x₁,y₁) = (1,2) and (x₂,y₂) = (-7,-4).

 

     (h,k) = ((-7-1)/2, (-4+2)/2) = (-4,-1)

 

So we have h = -4 and k = -1.  Plug those into the general equation of a circle:

 

     (x-(-4))² + (y-(-1))² = r²

     (x+4)² + (y+1)² = r²

 

To find the radius, r, use the distance formula from the center point, (-4,-1),1,2) to either endpoint, A or B.  I'll use point A, (1,2):

 

     r² = (x2-x1)² + (y2-y1)²

     r² = (-4-1)² + (-1-2)²

     r² = (-5)² + (-3)²

     r² = 25 + 9 = 36

 

So we have r² = 36 and we can write the final equation:

 

     (x+4)² + (y+1)² = 36

Answer 2

Find the midpoint of segment AB; that's the center; call those coordinates (h,k).

 

Find the length of AB and call half of that r.

 

Equation is (x-h)² + (y-k)^{2 = r}2 but you will put in the numbers h, k and r that you discover by following my instructions.