Equations of Circles Worksheet for 10th Grade | Lesson Planet - Free Printable

It looks like you've uploaded a worksheet about equations of circles, but the actual image isn't visible to me. However, based on your description and the text you provided, I can help you solve the problems listed in the worksheet.

Let’s go through each problem step by step and explain how to solve them.

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Problem 1:

Convert this equation into center-radius form:
$$
x^2 + y^2 - 6x + 8y - 11 = 0
$$

#### Step-by-step Solution:

We need to rewrite the equation in the standard form:
$$
(x - h)^2 + (y - k)^2 = r^2
$$
where $(h, k)$ is the center and $r$ is the radius.

Start by grouping $x$ and $y$ terms:
$$
(x^2 - 6x) + (y^2 + 8y) = 11
$$

Now complete the square for both $x$ and $y$:

- For $x^2 - 6x$: take half of $-6$, which is $-3$, square it: $9$
- For $y^2 + 8y$: take half of $8$, which is $4$, square it: $16$

Add these to both sides:
$$
(x^2 - 6x + 9) + (y^2 + 8y + 16) = 11 + 9 + 16
$$
$$
(x - 3)^2 + (y + 4)^2 = 36
$$

✔ Answer:
Center: $(3, -4)$, Radius: $\sqrt{36} = 6$
So, the center-radius form is:
$$
(x - 3)^2 + (y + 4)^2 = 36
$$

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Problem 2:

Write the standard equation of a circle with center at $(5, -1)$ and radius of 4.

Use the formula:
$$
(x - h)^2 + (y - k)^2 = r^2
$$
Plug in $h = 5$, $k = -1$, $r = 4$:
$$
(x - 5)^2 + (y + 1)^2 = 16
$$

✔ Answer:
$$
(x - 5)^2 + (y + 1)^2 = 16
$$

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Problem 3:

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

Again use the formula:
$$
(x + 2)^2 + (y - 3)^2 = 49
$$

✔ Answer:
$$
(x + 2)^2 + (y - 3)^2 = 49
$$

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Problem 4:

Write the standard equation of a circle with center at $(1, -7)$ and radius of 5.

$$
(x - 1)^2 + (y + 7)^2 = 25
$$

✔ Answer:
$$
(x - 1)^2 + (y + 7)^2 = 25
$$

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Problem 5:

Write the standard equation of a circle that is tangent to the x-axis, with a center located at $(3, 4)$.

If the circle is tangent to the x-axis, then the distance from the center to the x-axis equals the radius.

The y-coordinate of the center is 4 → so the radius is $|4| = 4$

Thus:
$$
(x - 3)^2 + (y - 4)^2 = 16
$$

✔ Answer:
$$
(x - 3)^2 + (y - 4)^2 = 16
$$

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Problem 6:

Sketch the graph of the circle given by the equation:
$$
(x - 2)^2 + (y + 3)^2 = 16
$$

#### Step-by-step:

- Center: $(2, -3)$
- Radius: $\sqrt{16} = 4$

To sketch:
1. Plot the center at $(2, -3)$
2. From the center, draw a circle with radius 4 in all directions
3. Mark points 4 units up, down, left, right:
- Up: $(2, 1)$
- Down: $(2, -7)$
- Left: $(-2, -3)$
- Right: $(6, -3)$

Then draw a smooth circle through those points.

✔ Graph Description: A circle centered at $(2, -3)$ with radius 4.

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Summary of Answers:

| Problem | Answer |
|--------|--------|
| 1 | $(x - 3)^2 + (y + 4)^2 = 36$ |
| 2 | $(x - 5)^2 + (y + 1)^2 = 16$ |
| 3 | $(x + 2)^2 + (y - 3)^2 = 49$ |
| 4 | $(x - 1)^2 + (y + 7)^2 = 25$ |
| 5 | $(x - 3)^2 + (y - 4)^2 = 16$ |
| 6 | Circle centered at $(2, -3)$, radius 4 |

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