# What is the equation of a circle with diameter AB; A is (5,4) and B is (-1,-4)?

First calculate the radius of the circle:

The horizontal distance between the points is the distance between the x coordinates of A and B, |5| + |-1| = 6, and the vertical distance is the distance between the y coordinates of A and B, |4| + |-4| = 8

The...

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First calculate the radius of the circle:

The horizontal distance between the points is the distance between the x coordinates of A and B, |5| + |-1| = 6, and the vertical distance is the distance between the y coordinates of A and B, |4| + |-4| = 8

The diameter is then the direct distance between A and B which is sqrt(6^2 + 8^2) = sqrt(36+64) = sqrt(100) = 10. This makes the radius 5.

We also need to know where the center of the circle is. This has coordinates halfway between the coordinates of A and B. Half the horizontal distance between A and B is 3 and half the vertical distance is 4, so that the center is at C(x,y) = B(x,y) + (3,4) = (-1,-4) + (3,4) = (2,0) (adding (3,4) again gives the coordinates of B).

The equation for a circle written in standard form is given by

`(x-x_c)^2 + (y-y_c)^2 = r^2`

where `r` is the radius and `(x_c,y_c)` are the coordinates of the center C.

Therefore the circle whose diameter is AB can be written in standard form as

`(x-2)^2 + y^2 = 25`