Equation of a Circle Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve the problems on this worksheet step by step and fill in the blanks as needed.

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Part 1: Determining if a Point is on a Circle

Formula:
$$
(x - h)^2 + (y - k)^2 = r^2
$$

To determine if a point is on a circle, substitute the point into the formula for x and y.

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#### Example 1: Is (2, 3) on the circle $(x - 2)^2 + (y + 1)^2 = 16$?

Substitute $x = 2$, $y = 3$:

$$
(2 - 2)^2 + (3 + 1)^2 = 0^2 + 4^2 = 0 + 16 = 16
$$

Since $16 = 16$, yes, the point (2, 3) is on the circle.

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#### Example 2: Is (0, 5) on the circle $(x + 1)^2 + (y - 4)^2 = 1$?

Substitute $x = 0$, $y = 5$:

$$
(0 + 1)^2 + (5 - 4)^2 = 1^2 + 1^2 = 1 + 1 = 2
$$

Since $2 \neq 1$, no, the point (0, 5) is not on the circle.

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#### Try: Is (5, 2) on the circle $(x + 1)^2 + (y - 2)^2 = 36$?

Substitute $x = 5$, $y = 2$:

$$
(5 + 1)^2 + (2 - 2)^2 = 6^2 + 0^2 = 36 + 0 = 36
$$

Since $36 = 36$, yes, the point (5, 2) is on the circle.

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Part 2: Writing the Equation Given the Center and a Point

Steps:
1. Substitute the center and point into the formula for $h$, $k$, $x$, and $y$.
2. Solve for the radius squared ($r^2$).
3. Write the equation with the center and $r^2$.

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#### Example: Write the equation of a circle with center (0, 0) and a point at (5, -12).

Center: $(h, k) = (0, 0)$
Point: $(x, y) = (5, -12)$

Use the formula:
$$
(x - h)^2 + (y - k)^2 = r^2
$$

Substitute:
$$
(5 - 0)^2 + (-12 - 0)^2 = r^2 \\
25 + 144 = r^2 \\
r^2 = 169
$$

So, the equation is:
$$
x^2 + y^2 = 169
$$

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#### Example: Write the equation of a circle with center (2, -3) and a point at (-4, 5).

Center: $(h, k) = (2, -3)$
Point: $(x, y) = (-4, 5)$

Plug into the formula:
$$
(-4 - 2)^2 + (5 - (-3))^2 = r^2 \\
(-6)^2 + (8)^2 = r^2 \\
36 + 64 = r^2 \\
r^2 = 100
$$

So, the equation is:
$$
(x - 2)^2 + (y + 3)^2 = 100
$$

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#### Try: Write the equation of a circle with center (-5, -7) and a point at (3, 0).

Center: $(-5, -7)$ → $h = -5$, $k = -7$
Point: $(3, 0)$ → $x = 3$, $y = 0$

Use:
$$
(3 - (-5))^2 + (0 - (-7))^2 = r^2 \\
(3 + 5)^2 + (0 + 7)^2 = r^2 \\
8^2 + 7^2 = r^2 \\
64 + 49 = r^2 \\
r^2 = 113
$$

So, the equation is:
$$
(x + 5)^2 + (y + 7)^2 = 113
$$

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✔ Final Answers Summary:

#### Determining if a point is on a circle:
- Fill in: x and y

#### Examples:
1. (2, 3): Yes
2. (0, 5): No
3. (5, 2): Yes

#### Writing the Equation:
- Steps:
1. Substitute
2. Solve for $r^2$
3. Write with center and $r^2$

#### Equations:
1. Center (0,0), point (5,-12):
$$ x^2 + y^2 = 169 $$

2. Center (2,-3), point (-4,5):
$$ (x - 2)^2 + (y + 3)^2 = 100 $$

3. Try: Center (-5,-7), point (3,0):
$$ (x + 5)^2 + (y + 7)^2 = 113 $$

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✔ All problems solved! Let me know if you'd like this formatted for printing or notes.