Exercises: Derive circle equation - Common Core Math - HS Geometry

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

What is the distance between the points $(1, 2)$ and $(4, 6)$ ?

A.

5

B.

7

C.

$\sqrt{7}$

D.

25

2.

A circle is defined as the set of all points in a plane that are:

A.

Inside a fixed boundary

B.

A fixed distance from a center point

C.

Connected by a curved line

D.

Equidistant from two fixed points

3.

Complete the square: $x^2 - 10x + $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $= (x - 5)^2$ . What number fills the blank?

B

Fluency Practice

1.

Write the equation of the circle with center $(3, -2)$ and radius $7$ in standard form.

2.

What is the center of the circle given by $(x + 4)^2 + (y - 1)^2 = 36$ ?

A.

$(4, 1)$

B.

$(-4, -1)$

C.

$(-4, 1)$

D.

$(4, -1)$

3.

What is the radius of the circle given by $(x - 5)^2 + (y + 3)^2 = 64$ ?

4.

Find the center and radius of the circle $x^2 + y^2 - 6x + 8y - 11 = 0$ . Enter the center as $(h, k)$ and the radius as $r$ .

5.

Convert $x^2 + y^2 + 10x - 4y + 20 = 0$ to standard form and identify the radius.

C

Varied Practice

1.

A circle has center $(-2, 5)$ and radius $3$ . Which equation represents this circle?

A.

$(x - 2)^2 + (y - 5)^2 = 3$

B.

$(x + 2)^2 + (y - 5)^2 = 9$

C.

$(x - 2)^2 + (y + 5)^2 = 9$

D.

$(x + 2)^2 + (y + 5)^2 = 3$

2.

For the circle $(x - 7)^2 + (y + 1)^2 = 100$ : the center is $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $ and the radius is $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $.

center (h, k):

radius:

3.

Complete the square to convert $x^2 + y^2 - 4x + 6y = 3$ to standard form.
Fill in the blanks: $(x^2 - 4x + \_\_\_) + (y^2 + 6y + \_\_\_) = 3 + \_\_\_ + \_\_\_$

value added to complete x-square:

value added to complete y-square:

value added to right side for x:

value added to right side for y:

4.

While completing the square on $x^2 + y^2 - 8x + 2y = 5$ , a student added 16 to the left side for the $x$ -group and 1 to the left side for the $y$ -group but wrote the right side as $5$ . What did the student forget to do?

A.

Divide both sides by 2 before completing the square

B.

Add 16 and 1 to the right side as well

C.

Take the square root of the right side

D.

Move the constant to the left side before grouping

5.

A point $(x, y)$ is on the circle with center $(h, k)$ and radius $r$ . Using the distance formula and the definition of a circle, explain in 2–3 sentences why the equation $(x - h)^2 + (y - k)^2 = r^2$ must be true for every such point.

D

Word Problems

1.

A circle passes through the point $(8, 3)$ and has its center at $(5, -1)$ .

Write the equation of this circle in standard form.

2.

A fountain in a city park is located at coordinates $(4, 6)$ on a map (measured in meters). The fountain sprays water that falls in a circular pattern with a radius of $\sqrt{45}$ meters.

1.

Write the equation of the boundary of the spray pattern in standard form.

2.

A bench is located at coordinates $(10, 9)$ . Is the bench within the spray pattern?

A.

Yes — the bench is strictly inside the spray boundary.

B.

No — the bench is outside the spray boundary.

C.

The bench is exactly on the spray boundary.

3.

A cell tower provides coverage in a circular area. The tower is at coordinates $(2, -3)$ (in kilometers) and the coverage area is described by the equation $x^2 + y^2 - 4x + 6y - 12 = 0$ .

1.

Complete the square to rewrite the equation in standard form, then identify the center and radius of the coverage area.

2.

A house at coordinates $(7, -3)$ wants to check if it receives coverage. Does it?

A.

Yes — the house is strictly inside the coverage area.

B.

No — the house is outside the coverage area.

C.

The house is exactly on the coverage boundary and receives edge coverage.

E

Error Analysis

1.

Priya is asked to identify the center and radius of $(x + 5)^2 + (y - 2)^2 = 49$ .

Priya's work:

Center: $(5, 2)$
Radius: $49$

Priya made two errors. Which choice correctly identifies both mistakes?

A.

The center should be $(-5, 2)$ and the radius should be $7$ .

B.

The center should be $(-5, -2)$ and the radius should be $7$ .

C.

The center should be $(5, 2)$ but the radius should be $7$ .

D.

The center should be $(-5, 2)$ and the radius should be $49$ .

2.

Marcus is converting $x^2 + y^2 - 6x + 4y = 12$ to standard form.

Marcus's work:

$(x^2 - 6x + 36) + (y^2 + 4y + 4) = 12 + 36 + 4$
$(x - 6)^2 + (y + 2)^2 = 52$
Center $(6, -2)$ , radius $\sqrt{52}$

Marcus made an error in Step 1 that affected Steps 2 and 3. Which choice correctly identifies the mistake and gives the right answer?

A.

He should have added $9$ (not $36$ ) for the $x$ -group, because half of $-6$ is $-3$ and $(-3)^2 = 9$ . The correct equation is $(x - 3)^2 + (y + 2)^2 = 25$ ; center $(3, -2)$ , radius $5$ .

B.

He used the correct value $36$ but forgot to add it to the right side. The equation should be $(x - 6)^2 + (y + 2)^2 = 12$ .

C.

He should have added $9$ for the $x$ -group and $16$ for the $y$ -group. The correct equation is $(x - 3)^2 + (y + 2)^2 = 37$ .

D.

The work is correct; the center is $(6, -2)$ and the radius is $\sqrt{52}$ .

F

Challenge / Extension

1.

Starting only from the definition of a circle and the Pythagorean Theorem (without using the distance formula), derive the equation $(x - h)^2 + (y - k)^2 = r^2$ for a circle with center $(h, k)$ and radius $r$ . Show your reasoning step by step.

2.