Circle Equation

This pattern asks you to identify the center and radius of a circle from its equation, or to build the equation given the center and radius.

 

Standard Form: Reading Center and Radius

The standard form of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

$(x + 2)^2 + (y - 5)^2 = 49$

What is the radius of this circle?

A) $7$
B) $2$
C) $5$
D) $49$

Since $r^2 = 49$, the radius is $r = 7$. The answer is A. Option D confuses $r^2$ with $r$. Options B and C are coordinates from the center, not the radius.

$(x + 5)^2 + (y - 2)^2 = 16$

What are the coordinates of the center?

A) $(5, -2)$
B) $(-5, 2)$
C) $(5, 2)$
D) $(-5, -2)$

Rewrite as $(x - (-5))^2 + (y - 2)^2 = 16$. The center is $(-5, 2)$. The answer is B. The signs flip: $(x + 5)$ means $h = -5$, and $(y - 2)$ means $k = 2$.

$(x - h)^2 + (y + 5)^2 = 121$, where $h$ is a constant. What is the center and radius?

A) Center $(-5, h)$, radius $11$
B) Center $(h, -5)$, radius $121$
C) Center $(h, -5)$, radius $11$
D) Center $(-5, h)$, radius $121$

$(y + 5) = (y - (-5))$, so $k = -5$. The center is $(h, -5)$ and $r = \sqrt{121} = 11$. The answer is C.

Key rule: The signs in the equation are opposite to the coordinates of the center. $(x + a)$ means the $x$-coordinate of the center is $-a$.

 

Building the Equation from Center and Radius

Plug $(h, k)$ and $r$ into the standard form. Remember to square the radius.

A circle has center $(-5, 12)$ and radius $4p$. Which equation represents the circle?

A) $(x + 5)^2 + (y - 12)^2 = 4p$
B) $(x + 5)^2 + (y - 12)^2 = 16p$
C) $(x + 5)^2 + (y - 12)^2 = 4p^2$
D) $(x + 5)^2 + (y - 12)^2 = 16p^2$

Center $(-5, 12)$ gives $(x + 5)^2 + (y - 12)^2$. Radius $= 4p$, so $r^2 = (4p)^2 = 16p^2$. The answer is D. Option A forgets to square the radius entirely. Option C only squares $p$ but not 4.

 

Completing the Square

When the equation is in expanded form like $x^2 + 10x + y^2 = 16y + 12$, rewrite it in standard form by completing the square.

$x^2 + 10x + y^2 = 16y + 12$

What is the radius?

A) $12$
B) $\sqrt{51}$
C) $\sqrt{101}$
D) $\sqrt{368}$

Rearrange: $x^2 + 10x + y^2 - 16y = 12$.
Complete the square for $x$: $x^2 + 10x + 25 = (x + 5)^2$. Add 25 to both sides.
Complete the square for $y$: $y^2 - 16y + 64 = (y - 8)^2$. Add 64 to both sides.
$(x + 5)^2 + (y - 8)^2 = 12 + 25 + 64 = 101$.
$r = \sqrt{101}$. The answer is C.

$x^2 + y^2 - 36y = 0$

What is the center?

A) $(18, 0)$
B) $(0, 18)$
C) $(-18, 0)$
D) $(0, -18)$

Complete the square for $y$: $x^2 + (y^2 - 36y + 324) = 324$, giving $x^2 + (y - 18)^2 = 324$. Center is $(0, 18)$. The answer is B.

Completing the square recipe: For $x^2 + bx$, add $\left(\dfrac{b}{2}\right)^2$ to both sides to get $\left(x + \dfrac{b}{2}\right)^2$.

 

Diameter from Radius

Some problems give the standard form and ask for the diameter. Remember: diameter $= 2r$.

$(x + 6)^2 + (y - 1)^2 = 49$. What is the diameter?

A) $7$
B) $14$
C) $49$
D) $98$

$r = 7$. Diameter $= 2(7) = 14$. The answer is B.

 

Shifting a Circle

Translating a circle right by $a$ units and up by $b$ units changes the center from $(h, k)$ to $(h + a, k + b)$. The radius stays the same unless the problem says otherwise.

$(x + 3)^2 + y^2 = 25$ defines circle P. Circle Q is P shifted right by 5 units. What is the equation of Q?

A) $(x + 8)^2 + y^2 = 25$
B) $(x + 3)^2 + (y - 5)^2 = 25$
C) $(x - 2)^2 + y^2 = 25$
D) $(x + 3)^2 + (y + 5)^2 = 25$

P has center $(-3, 0)$. Shift right 5: new center $(-3 + 5, 0) = (2, 0)$. Equation: $(x - 2)^2 + y^2 = 25$. The answer is C. Option A shifts left instead of right. Options B and D shift vertically.

Circle P: $(x - 5)^2 + (y + 1)^2 = 4$. Circle Q is P translated right 3 and up 4, with radius 3 times as large. What is Q's equation?

A) $(x - 8)^2 + (y + 5)^2 = 36$
B) $(x - 8)^2 + (y - 3)^2 = 36$
C) $(x - 8)^2 + (y - 3)^2 = 12$
D) $(x - 2)^2 + (y + 5)^2 = 12$

P center: $(5, -1)$, radius $= 2$. New center: $(5 + 3, -1 + 4) = (8, 3)$. New radius $= 3 \times 2 = 6$, so $r^2 = 36$. The answer is B.

 

Expanded Form: Finding a Coefficient

Some problems give center and radius and ask for a specific coefficient when the equation is expanded to $x^2 + y^2 + ax + by + c = 0$.

A circle has center $(4, -7)$ and radius 6. The equation is $x^2 + y^2 + ax + by + c = 0$. What is $a$?

Standard form: $(x - 4)^2 + (y + 7)^2 = 36$. Expand: $x^2 - 8x + 16 + y^2 + 14y + 49 = 36$. Rearrange: $x^2 + y^2 - 8x + 14y + 29 = 0$. So $a = -8$.

Same circle. What is $f$ if the equation is $x^2 + y^2 + dx + ey + f = 0$?

From above: $f = 29$. But if radius is 8 instead: $(x - 4)^2 + (y + 7)^2 = 64$. Expand: $x^2 - 8x + 16 + y^2 + 14y + 49 = 64$. So $x^2 + y^2 - 8x + 14y + 1 = 0$ and $f = 1$.

Tip: To expand, use $(x - h)^2 = x^2 - 2hx + h^2$. Then move $r^2$ to the left side so the equation equals zero.

 

What to Do on Test Day

• Standard form: $(x - h)^2 + (y - k)^2 = r^2$. Center is $(h, k)$, radius is $r$.
• Signs flip: $(x + 5)$ means $h = -5$. $(y - 3)$ means $k = 3$. The signs in the equation are the opposite of the center coordinates.
• $r^2$ vs. $r$: The right side of the equation is $r^2$, not $r$. If the equation says $= 49$, the radius is $7$, not $49$. This is the most common wrong answer.
• Completing the square recipe: For $x^2 + bx$, add $\left(\dfrac{b}{2}\right)^2$ to both sides. Don't forget to add the same constant to both sides.
• Diameter = $2r$. If they ask for the diameter, double the radius.
• Shifting a circle: Moving right by $a$ changes the center's $x$-coordinate by $+a$. Moving up by $b$ changes $y$ by $+b$. The radius stays the same unless stated otherwise.
• Expanded form: To find a coefficient in $x^2 + y^2 + ax + by + c = 0$, expand the standard form and collect terms.