Given two points, A(-2,5) and B(-4,3) find the equation of the circle with diameter AB.?

A(-2,5) and B(-4,3) :

midpoint:
x = (-2 - 4)/2 = -3
y = (5 + 3)/2 = 4

=> centre: (-3, 4)

distance, AB:
d = √((-2 - (-4))^2 + (5 - 3)^2)
d = √8 = 2√2

=> radius = 1/2 * 2√2 = √2

(x - (-3))^2 + (y - 4)^2 = √2^2
(x + 3)^2 + (y - 4)^2 = 2

https://www.desmos.com/calculator/g5nktouem9

Learn this method. It is often useful, and there is no need to derive either the center coordinates or the radius. The resulting equation is in general form.

Circle with diameter AB, A(-2, 5) and B(-4, 3):

(x + 2)(x + 4) + (y - 5)(y - 3) = 0
x² + y² + 6x - 8y + 23 = 0

Now, here is why it works.
Let P(x, y) be any point on the circle where x ≠ -2 and x ≠ -4.

slope of PA = (y - 5)/(x + 2)
slope of PB = (y - 3)/(x + 4)

The angle formed by chords PA and PB intercepts a diameter, and is therefore a right angle. It follows that PA and PB are perpendicular, so the product of their slopes is -1.

[(y - 5)/(x + 2)][(y - 3)/(x + 4)] = -1
(y - 5)(y - 3) = -(x + 2)(x + 4)
(x + 2)(x + 4) + (y - 5)(y - 3) = 0

The center of the circle is the midpoint of [AB]

xC = (xA + xB)/2 = (- 2 - 4)/2 = - 6/2 = - 3

yC = (yA + yB)/2 = (5 + 3)/2 = 8/2 = 4


The distance [AB] is the diameter

xAB = xB - xA = - 4 - (- 2) = - 4 + 2 = - 2

yAB = yB - yA = 3 - 5 = - 2

AB² = xAB² + yAB²

AB² = (- 2)² + (- 2)²

AB² = 8

AB = 2√2 ← this is the diameter, so to obtain the radius, you divide by 2

R = √2 ← this is the radius of the circle


The typical equation of a circle is: (x - xo)² + (y - yo)² = R² → where:

xo: abscissa of center → (- 3) in your case

yo: ordinate of center → (4) in your case

R: radius of circle → √2 in your case


→ The equation of the circle is: (x + 3)² + (y - 4)² = 2

C ( - 3 , 4 )
r² = (-4 + 3)² + (3 - 4)²
r² = 1 + 1 = 2
( x + 3)² + (y - 4)² = 2

midpoint AB = center of the circle
C = (-3, 4)
radius is dist. from center to either A or B
r = sqrt(1 + 1) so r^2 = 2

(x + 3)^2 + (y - 4)^2 = 2 <<< answer

You need to find the mid-point of AB , which is ( - 3, 4) . This is the circle centre.

Hence equation is
(x - - 3)^2 + ( y - 4)^2 = 5^2
x^2 + 6x + 9 + y^2 - 8y + 16 = 25
x^2 + y^2 + 6x - 8y + 9 + 16 - 25 = 0
x^2 + y^2 + 6x - 8y = 0

Centre (-3, 4) and radius squared is 1^2 + 1^2
(x + 3)^2 + (y - 4)^2 = 2