General To Standard Form Of A Circle Worksheet - Fill Online ... - Free Printable

Problem Overview:

The task is to write the equation of each circle in standard form, identify the center of the circle, and determine its radius. The given equations are in general form, which is:

\[
x^2 + y^2 + Dx + Ey + F = 0
\]

The standard form of a circle's equation is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

To convert from general form to standard form, we use the method of completing the square for both \(x\) and \(y\).

---

Step-by-Step Solution:

#### 1. \( x^2 + y^2 + 8x - 6y + 23 = 35 \)

1. Rearrange terms:
\[
x^2 + 8x + y^2 - 6y = 35 - 23
\]
\[
x^2 + 8x + y^2 - 6y = 12
\]

2. Complete the square for \(x\):
- Take the coefficient of \(x\), which is 8, divide by 2 to get 4, and square it to get \(4^2 = 16\).
- Add and subtract 16 inside the equation:
\[
x^2 + 8x + 16 - 16
\]
- This becomes:
\[
(x + 4)^2 - 16
\]

3. Complete the square for \(y\):
- Take the coefficient of \(y\), which is -6, divide by 2 to get -3, and square it to get \((-3)^2 = 9\).
- Add and subtract 9 inside the equation:
\[
y^2 - 6y + 9 - 9
\]
- This becomes:
\[
(y - 3)^2 - 9
\]

4. Substitute back into the equation:
\[
(x + 4)^2 - 16 + (y - 3)^2 - 9 = 12
\]

5. Simplify:
\[
(x + 4)^2 + (y - 3)^2 - 25 = 12
\]
\[
(x + 4)^2 + (y - 3)^2 = 37
\]

6. Identify the center and radius:
- Center: \((-4, 3)\)
- Radius: \(\sqrt{37}\)

Standard form: \((x + 4)^2 + (y - 3)^2 = 37\)

---

#### 2. \( x^2 + y^2 + 8x + 15 = 3 \)

1. Rearrange terms:
\[
x^2 + 8x + y^2 = 3 - 15
\]
\[
x^2 + 8x + y^2 = -12
\]

2. Complete the square for \(x\):
- Coefficient of \(x\) is 8, so \(\left(\frac{8}{2}\right)^2 = 16\).
- Add and subtract 16:
\[
x^2 + 8x + 16 - 16 + y^2 = -12
\]
\[
(x + 4)^2 - 16 + y^2 = -12
\]

3. Simplify:
\[
(x + 4)^2 + y^2 = 4
\]

4. Identify the center and radius:
- Center: \((-4, 0)\)
- Radius: \(\sqrt{4} = 2\)

Standard form: \((x + 4)^2 + y^2 = 4\)

---

#### 3. \( x^2 + y^2 + 6x + 6 = 9/18 \)

1. Simplify the constant term on the right:
\[
9/18 = 0.5
\]
So the equation becomes:
\[
x^2 + y^2 + 6x + 6 = 0.5
\]

2. Rearrange terms:
\[
x^2 + 6x + y^2 = 0.5 - 6
\]
\[
x^2 + 6x + y^2 = -5.5
\]

3. Complete the square for \(x\):
- Coefficient of \(x\) is 6, so \(\left(\frac{6}{2}\right)^2 = 9\).
- Add and subtract 9:
\[
x^2 + 6x + 9 - 9 + y^2 = -5.5
\]
\[
(x + 3)^2 - 9 + y^2 = -5.5
\]

4. Simplify:
\[
(x + 3)^2 + y^2 = 3.5
\]

5. Identify the center and radius:
- Center: \((-3, 0)\)
- Radius: \(\sqrt{3.5}\)

Standard form: \((x + 3)^2 + y^2 = 3.5\)

---

#### 4. \( x^2 + y^2 - 8x - 9y + 15 = 27 \)

1. Rearrange terms:
\[
x^2 - 8x + y^2 - 9y = 27 - 15
\]
\[
x^2 - 8x + y^2 - 9y = 12
\]

2. Complete the square for \(x\):
- Coefficient of \(x\) is -8, so \(\left(\frac{-8}{2}\right)^2 = 16\).
- Add and subtract 16:
\[
x^2 - 8x + 16 - 16 + y^2 - 9y = 12
\]
\[
(x - 4)^2 - 16 + y^2 - 9y = 12
\]

3. Complete the square for \(y\):
- Coefficient of \(y\) is -9, so \(\left(\frac{-9}{2}\right)^2 = \left(\frac{9}{2}\right)^2 = \frac{81}{4}\).
- Add and subtract \(\frac{81}{4}\):
\[
(x - 4)^2 - 16 + y^2 - 9y + \frac{81}{4} - \frac{81}{4} = 12
\]
\[
(x - 4)^2 + \left(y - \frac{9}{2}\right)^2 - 16 - \frac{81}{4} = 12
\]

4. Simplify:
- Convert 16 to a fraction with denominator 4:
\[
16 = \frac{64}{4}
\]
- Combine constants:
\[
-16 - \frac{81}{4} = -\frac{64}{4} - \frac{81}{4} = -\frac{145}{4}
\]
- Add \(\frac{145}{4}\) to both sides:
\[
(x - 4)^2 + \left(y - \frac{9}{2}\right)^2 = 12 + \frac{145}{4}
\]
\[
12 = \frac{48}{4}
\]
\[
\frac{48}{4} + \frac{145}{4} = \frac{193}{4}
\]

5. Identify the center and radius:
- Center: \((4, \frac{9}{2})\)
- Radius: \(\sqrt{\frac{193}{4}} = \frac{\sqrt{193}}{2}\)

Standard form: \((x - 4)^2 + \left(y - \frac{9}{2}\right)^2 = \frac{193}{4}\)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ (x + 4)^2 + (y - 3)^2 = 37, \text{ Center: } (-4, 3), \text{ Radius: } \sqrt{37} \\
2. & \ (x + 4)^2 + y^2 = 4, \text{ Center: } (-4, 0), \text{ Radius: } 2 \\
3. & \ (x + 3)^2 + y^2 = 3.5, \text{ Center: } (-3, 0), \text{ Radius: } \sqrt{3.5} \\
4. & \ (x - 4)^2 + \left(y - \frac{9}{2}\right)^2 = \frac{193}{4}, \text{ Center: } \left(4, \frac{9}{2}\right), \text{ Radius: } \frac{\sqrt{193}}{2}
\end{aligned}
}
\]