How do you write the equation of the circle with diameter that has endpoints at (–4, –1) and (–8, –9)?

Question

4571 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-06-23T09:42:57

#(x+6)^2+(y+5)^2=20#

To write the equation of a circle we need to know the center and the radius. Since we know two endpoints, we know the extrema of a diameter.

The idea is the following: we will find the center as the midpoint of the diameter, and the radius as half the length of the diameter.

To find the midpoint of a segmente with extrema #A=(x_A,y_A)# and #B=(x_B,y_B)#, we have the following formula:

#M = (\frac{x_A+x_B}{2}, \frac{y_A+y_B}{2})#

In your case, this becomes

#C = (\frac{-4-8}{2}, \frac{-1-9}{2})=(-6,-5)#

As for the length, we have the following formula:

#d = sqrt((x_A-x_B)^2+(y_A-y_B)^2)#

Which in your case becomes

#d = sqrt((-4+8)^2+(-1+9)^2)=sqrt(16+64)=sqrt(80) = 4sqrt(5)#

So, the diameter is #4sqrt(5)# units long, which means that the radius is #2sqrt(5)# units long.

Now we can write the equation:

in general, given the center #(x_0,y_0)# and the radius #r#, we have

#(x-x_0)^2+(y-y_0)^2=r^2#

Which in this case becomes

#(x+6)^2+(y+5)^2=20#