Question

How do you write an equation for a circle with (-10 , 0) to (-16 , -10) as a diameter?

Answer 1

The equation is `(x + 13)^2 + (y + 5)^2 = 34`

Explanation:

Start by finding the length of the diameter using the distance formula.

`d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

`d = sqrt((-6)^2 + (-10)^2)`

`d = sqrt(136)`

`d = 2sqrt(34)`

We will now find the radius using the formula `d = 2r`

`2sqrt(34) = 2r`

`sqrt(34) = r`

Now, let's find the centre of the circle. We can do this using the midpoint formula.

Let `c` be the centre:

`c = ((x_1 + x_2)/2, (y_1 + y_2)/2)`

`c = ((-10 + (-16))/2, (-10 + 0)/2)`

`c = (-26/2, -10/2)`

`c = (-13, -5)`

We can now write the equation of the circle. The form `(x - a)^2 + (x- b)^2 = r^2` represents the equation of the circle, where the point `(a, b)` is the centre and `r` is the radius.

Substituting, we get:

`(x - (-13))^2 + (y - (-5))^2 = (sqrt(34))^2`

`(x + 13)^2 + (y + 5)^2 = 34`

Hopefully this helps!