Question Video: Finding the Center and Radius of a Circle from Its Equation Mathematics • 11th Grade

Find the center and radius of the circle (𝑥 − 2)² + (𝑦 + 8)² − 100 = 0.

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Video Transcript

Find the centre and radius of the circle 𝑥 minus two squared plus 𝑦 plus eight squared minus 100 equals zero.

To answer this question, we can recall the centre radius form of the equation of a circle. If a circle has centre with coordinates ℎ, 𝑘 and a radius of 𝑟 units, then its equation can be written in centre radius form: 𝑥 minus ℎ squared plus 𝑦 minus 𝑘 squared equals 𝑟 squared.

We can see that the equation we’ve been given is very nearly in this form. But it has a negative 100 on the left and a zero on the right. So we need to rearrange our equation slightly by adding 100 to both sides. This cancels out the negative 100 on the left, and now we have positive 100 on the right. So our equation has become 𝑥 minus two squared plus 𝑦 plus eight squared is equal to 100.

We can now compare these two equations. Firstly, on the right of the equation, we see that 𝑟 squared is equal to 100. To find the value of 𝑟, we need to take the square root of both sides of this equation. 𝑟 is equal to the square root of 100, which is 10. This tells us then that the radius of the circle we’ve been given is 10 units.

Now let’s consider its centre. Comparing the first bracket in each equation — so that’s the bracket with the 𝑥 in — we can see that ℎ is equal to two, which means that the 𝑥-coordinate of the centre is two. Now let’s compare the second brackets, and this is slightly more tricky. In the general form, we have negative 𝑘, but in our circle we have positive eight. So we have that negative 𝑘 is equal to eight.

To find the value of 𝑘, we must either divide or multiply both sides of this equation by negative one, and it gives that 𝑘 is equal to negative eight. We can see that if we take 𝑦 and we subtract negative eight — that’s our value of 𝑘 — then the two negative signs next to each other form a positive overall. So 𝑦 minus negative eight is equal to 𝑦 plus eight, which is the expression we have in our circle. So the centre of this circle is the point with coordinates two, negative eight. And as we’ve already found that the radius is 10 units, we’ve answered the question.

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