Video Transcript
Find the center and radius of the
circle 𝑥 plus four squared plus 𝑦 minus two squared is equal to 225.
We’re given the equation for a
circle. We need to use this to find the
center of our circle and the radius of our circle. To start, let’s recall the equation
of a circle. We know a circle with a center at
the point ℎ, 𝑘 and a radius of 𝑟 will have the equation 𝑥 minus ℎ all squared
plus one minus 𝑘 all squared is equal to 𝑟 squared. And we can see the equation we’re
given is almost in this form. We do have to be careful,
however. For example, we’re not subtracting
a constant from 𝑥; we’re adding the constant four. But remember, adding four is the
same as subtracting negative four. So we can in fact write this as 𝑥
minus negative four all squared plus 𝑦 minus two all squared is equal to 225.
Now, it’s really easy to see the
center of our circle. Our value of ℎ is negative four,
and our value of 𝑘 is two. All we have to do now is find the
radius of our circle. In this case, the radius squared
will be equal to 225. So we want 𝑟 squared is equal to
225. There’s a few different ways of
doing this. For example, we could take the
square roots of both sides of this equation. Normally, we would get a positive
and a negative square root. But remember, in this case, this
represents the radius. This is a length, so it must be
positive. So we get that 𝑟 is equal to the
positive square root of 225. We can calculate this; it’s equal
to 15. So we can write 225 as 15
squared. This means the radius of our circle
must be equal to 15.
Remember, the center of our circle
will be the point ℎ, 𝑘. We’ve shown that ℎ is equal to
negative four and 𝑘 is equal to two. And of course, we already showed
the radius was 15. Therefore, given the equation of
the circle 𝑥 plus four all squared plus 𝑦 minus two all squared is equal to 225,
we were able to show the center of this circle was the point negative four, two and
the radius of this circle was 15.
