Question

1) If the endpoints of the diameter of a circle are (9, 4) and (5, 2), what is the standard form equation of the circle?

Answer 1

First thing you need to do is find the center. This would be the midpoint of the two points given. 

Midpoint = $( \frac{X1+X2}{2} , \frac{Y1+Y2}{2} )$
where X1 is the x-coordinate of the first ordered pair and Y1 is the y-coordinate of the first ordered pair. Same applies to X2 and Y2. So now we plug in.

Mdpt = $( \frac{9+5}{2} , \frac{4+2}{2} )$
= $( \frac{14}{2} , \frac{6}{2} )$

Your midpoint is at (7, 3).

This gives you the beginning of the equation. The equation of a circle is:

(x - h)² + (y - k)² = radius²
where (h, k) is the center. 

Your equation is:
(x - 7)² + (y - 3)² = radius²

To find the radius, we do either the distance formula from the center to either of the endpoints. Or the distance between the two endpoints divided by two. Depends on which one you want to do. I choose the latter, but it's all personal preference.

Distance formula = √((X2 - X1)² + (Y2 - Y1)²)

So we plug in:

D = √((5 - 9)² + (2 - 4)²)
D = √((- 4)² + (- 2)²)
D = √(16 + 4)
D = √20

Distance of the diameter = 2√5

Now divide that by 2, your radius is √5

Now we plug in to the equation:

(x - 7)² + (y - 3)² = (√5)²

Your final equation is:

(x - 7)² + (y - 3)² = 5