Video: Finding the Center and Radius of a Circle from Its Equation

Find the center and radius of the circle (π‘₯ + 4)Β² + (𝑦 βˆ’ 2)Β² = 225.

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Video Transcript

Find the center and radius of the circle π‘₯ 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 β„Ž, π‘˜ and a radius of π‘Ÿ will have the equation π‘₯ minus β„Ž all squared plus one minus π‘˜ all squared is equal to π‘Ÿ 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 π‘₯; 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 π‘₯ 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 β„Ž is negative four, and our value of π‘˜ 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 π‘Ÿ 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 π‘Ÿ 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 β„Ž, π‘˜. We’ve shown that β„Ž is equal to negative four and π‘˜ is equal to two. And of course, we already showed the radius was 15. Therefore, given the equation of the circle π‘₯ 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.

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