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.
