Circles Pattern - Applying Equations

This pattern asks you to use a circle's equation to check if a point lies on it, find missing coordinates, or determine possible values for points on the circle.

Finding Possible $y$-Values for a Point on the Circle

A point $(a, b)$ lies on the circle $(x - h)^2 + (y - k)^2 = r^2$ only if $b$ is within the range $[k - r, k + r]$.

$(x - 5)^2 + (y + 8)^2 = 81$

If $(a, b)$ lies on this circle, which is a possible value for $b$?

A) $-18$
B) $-15$
C) $5$
D) $9$

Center is $(5, -8)$ and $r = 9$. The $y$-values range from $-8 - 9 = -17$ to $-8 + 9 = 1$. Check each option: $-18 < -17$ (out), $-15$ is in $[-17, 1]$ (valid), $5 > 1$ (out), $9 > 1$ (out). The answer is B.

$(x - 5)^2 + (y + 8)^2 = 144$

If $(m, n)$ lies on this circle, which is a possible value for $n$?

A) $-21$
B) $5$
C) $12$
D) $-19$

Center $(5, -8)$, $r = 12$. Range: $[-20, 4]$. Only $-19$ falls in this range. The answer is D.

Quick check: Find the center's $y$-coordinate and the radius. The valid range is $[k - r, k + r]$. Eliminate options outside this range.

Substituting to Find a Missing Coordinate

If a point with one known coordinate lies on the circle, substitute it and solve for the other.

$(x + 2)^2 + (y - 4)^2 = 16$. The point $(k, 8)$ lies on the circle. What is $k$?

Substitute $y = 8$: $(k + 2)^2 + (8 - 4)^2 = 16$, so $(k + 2)^2 + 16 = 16$. Then $(k + 2)^2 = 0$, giving $k = -2$.

$(x - 3)^2 + (y + 1)^2 = 25$. The point $(7, b)$ lies on the circle. What is $b$?

$(7 - 3)^2 + (b + 1)^2 = 25$, so $16 + (b + 1)^2 = 25$. Then $(b + 1)^2 = 9$, giving $b + 1 = \pm 3$, so $b = 2$ or $b = -4$.

When the equation yields two solutions, check which one the problem asks for (some problems say "positive value" or give specific answer options).

Diameter Endpoints to Equation

Given the endpoints of a diameter, the center is the midpoint and the radius is half the distance between them.

A circle has a diameter with endpoints $(-1, -6)$ and $(11, -6)$. The equation is $(x - 5)^2 + (y + 6)^2 = r^2$. What is $r$?

Center $= \left(\dfrac{-1 + 11}{2}, \dfrac{-6 + (-6)}{2}\right) = (5, -6)$. Diameter $= 11 - (-1) = 12$, so $r = 6$.

A circle has a diameter with endpoints $(-3, 5)$ and $(11, 5)$. What is $r$?

Center $= (4, 5)$. Diameter $= 11 - (-3) = 14$, so $r = 7$.

Scaling a Circle's Radius

If circle Q has the same center as circle P but a radius $n$ times as large, multiply $r$ by $n$ and square to get the new $r^2$.

Circle P: $(x - 3)^2 + (y + 7)^2 = 36$. Circle Q has the same center and radius 3 times as large. The equation of Q is $(x - 3)^2 + (y + 7)^2 = n$. What is $n$?

P has $r = 6$. Q has $r = 18$. So $n = 18^2 = 324$.

Verifying a Point on the Circle

To check if a specific point lies on the circle, substitute and see if both sides are equal.

Does $(1, 2)$ lie on the circle $(x - 4)^2 + (y + 1)^2 = 18$?

$(1 - 4)^2 + (2 + 1)^2 = 9 + 9 = 18$. Yes — it equals $r^2$, so the point is on the circle.

What to Do on Test Day

• Point on the circle test: Substitute the point's coordinates into the equation. If both sides are equal, the point is on the circle.
• Possible $y$-values: For center $(h, k)$ and radius $r$, the $y$-coordinate of any point on the circle must be in the range $[k - r, k + r]$. Eliminate any answer choice outside this range.
• Finding a missing coordinate: Substitute the known coordinate, solve the resulting equation. You may get two solutions — check which one the problem asks for.
• Diameter endpoints → equation: Center = midpoint. Radius = half the distance between endpoints. Then plug into $(x - h)^2 + (y - k)^2 = r^2$.
• Scaling radius: If the new radius is $n$ times the old, the new $r^2$ is $n^2$ times the old $r^2$. Don't forget to square.