Video: Finding the Equation of a Circle

Video Transcript

What is the equation of the circle of radius 24 that lies in the third quadrant and is tangent to the two axes?

A coordinate grid is divided into four quadrants by its axes. The third quadrant is the area where both 𝑥 and 𝑦 are negative. We’re told that this circle lies completely in the third quadrant and is tangent to the two axes, which means that the two axes both touch the circumference of the circle but don’t cross inside the circle. So the circle looks something like this.

We’re also told that the circle has a radius of 24. This means that the vertical distance from the center of the circle to the 𝑥-axis is 24 units. And the horizontal distance from the center of the circle to the 𝑦-axis is also 24 units. The circle therefore meets both the 𝑥- and 𝑦-axes at negative 24. So now we know what our circle looks like. But let’s think about how to find its equation. The standard format for writing the equation of a circle is center-radius form. If a circle has a center at the point ℎ𝑘 and a radius of 𝑟 units, then its equation in center-radius form is 𝑥 minus ℎ squared plus 𝑦 minus 𝑘 squared equals 𝑟 squared.

We know the radius of our circle; it’s 24. And, from the sketch we’ve drawn, we can also work out its center. The center of our circle is the point negative 24, negative 24. So substituting each of these values into the equation of our circle gives 𝑥 minus negative 24 squared plus 𝑦 minus negative 24 squared equals 24 squared. We can simplify each of the brackets as the two negative signs next to each other combined to form a positive sign. 𝑥 minus negative 24 is the same as 𝑥 plus 24. So we have 𝑥 plus 24 squared plus 𝑦 plus 24 squared equals 24 squared.

Now, we could leave our answer in center-radius form. But, in this question, we’re going to expand the brackets and give our answer in expanded form instead. Let’s just look at one of the brackets first of all, 𝑥 plus 24 squared, which, remember, means 𝑥 plus 24 multiplied by 𝑥 plus 24. If we expand these brackets, perhaps using the FOIL method, we get 𝑥 squared plus 24𝑥 plus 24𝑥 plus 576. Grouping the like terms in the middle gives 𝑥 squared plus 48𝑥 plus 576.

Now, the expansion for 𝑦 will be exactly the same. But instead of 𝑥s, we’ll have 𝑦s. So the equation of our circle becomes 𝑥 squared plus 48𝑥 plus 576 plus 𝑦 squared plus 48𝑦 plus 576 equals 576. Now, one lot of 576 on the left will cancel with the 576 on the right. And it’s usual when given the equation of a circle in expanded form to give the terms in the order 𝑥 squared, 𝑦 squared, 𝑥, 𝑦, and then the constant term. So rearranging the order of the terms gives the equation of this circle in expanded form. 𝑥 squared plus 𝑦 squared plus 48𝑥 plus 48𝑦 plus 576 equals zero.