Points (6, –7) and (8, –9) are endpoints of the diameter of a circle.

Question

Points (6, –7) and (8, –9)  are endpoints of the diameter of a circle.

 

(a) What is the length of the diameter? Give the exact answer, simplified as much as possible.

(b) What is the center of the circle?

(c) What is the equation of the circle?

Answer 1

To find the length of the circle's diameter, we use the distance formula.

 

d = √[(x₂ - x₁)² + (y₂ - y₁)²]           

 

where:

Point 1 is (6, -7)

Point 2 is (8, -9)

 

 

d = √[(8 - 6)² + (-9 - (-7))²]

 

  = √[(2)² + (-2)²]

 

  = √(4 + 4)

  = √(8)

  = √(4) √(2)

  = 2 √2

 

The length of the diameter is 2√2.

 

 

To find the center of the circle, we take the midpoint using the two endpoints.  This is because the midpoint is the center of a line segment, being the diameter.

 

M = (    (x₁ + x₂) / 2     ,     (y₁ + y₂) / 2   )

 

   = ( (6 + 8) / 2   ,    (-7 - 9) / 2 )

   = (14/2 , -16/2)

   = (7 , -8)

 

The center of the circle is  (7, -8).

 

 

The find the equation of the circle, we use the diameter length, and center point.
 
The general equation of a circle is
 
x² + y² = r²
 
where:
r = radius
 
and the centerpoint of the circle is located at the origin (0, 0).

 

 

Since our centerpoint is located at (7, -8)  and the radius is  √2,  our equation of the circle is

 

 

(x - 7)² + (y + 8)² = 2