#14 · Circle radius? Don't trust the 42.
The question. The graph of this equation in the xy-plane is a circle. What is its radius?
The trap. 42 is sitting on the right, so the radius looks like √42. It isn't. A circle's formula is two perfect squares equal to r²:
Here (h, k) is the center and r is the radius. The number on the right is r² only once the left side has that shape, and ours doesn't yet.
Complete the square twice. First get the x terms and the y terms together on the left. Moving 6y over gives:
Now find the number that turns each part into a perfect square: take half of the coefficient, then square it.
- For x² + 14x: half of 14 is 7, so add 7² = 49.
- For y² − 6y: half of −6 is −3, so add (−3)² = 9.
Add both amounts to both sides of the equation, then write the left side as two perfect squares:
Now the equation is in the circle's form, so you can read r² from the right side: r² = 100 and the radius is 10. The center is (−7, 3).
The shortcut: Desmos
Type the equation just as it's written, with no rearranging, and zoom out, since the default view cuts off the top of the circle. Click the circle, then click its top and bottom points. Desmos labels them (−7, 13) and (−7, −7). The full height is the diameter, 13 − (−7) = 20, so divide by 2: the radius is 10.
If the radius isn't a whole number, the labels are rounded decimals, so the graph gives you an estimate. Change the 42 to 43 and they read (−7, 13.04988) and (−7, −7.04988), so the radius is about 10.05. Match that to the answer choices, or complete the square for the exact value: r² = 101, so r = √101.
If you only need the radius
You can skip writing the left side as squares. Once you have the numbers you'd add, the right side is all that matters: r² = 42 + 49 + 9 = 100. For an equation arranged as x² + y² + Dx + Ey = F, that's:
The takeaway: don't trust the number on the right until the equation is in the circle's form. Then you can read r² straight from the right side. Or let Desmos skip the algebra and read the circle's top and bottom.