Find the​ x- and​ y- intercepts of the graph of the given equation.
left parenthesis x minus 4 right parenthesis squared plus left parenthesis y minus 9 right parenthesis squared equals 81


The given equation is:

\[
(x - 4)^2 + (y - 9)^2 = 81
\]

This represents the equation of a circle with a center at \((4, 9)\) and a radius \(r = \sqrt{81} = 9\).

### Finding the Intercepts

#### 1. **X-intercepts**:
To find the x-intercepts, set \(y = 0\) and solve for \(x\).

Substitute \(y = 0\) into the equation:

\[
(x - 4)^2 + (0 - 9)^2 = 81
\]

This simplifies to:

\[
(x - 4)^2 + 81 = 81
\]

\[
(x - 4)^2 = 0
\]

\[
x - 4 = 0
\]

\[
x = 4
\]

Thus, the x-intercept is \((4, 0)\).

#### 2. **Y-intercepts**:
To find the y-intercepts, set \(x = 0\) and solve for \(y\).

Substitute \(x = 0\) into the equation:

\[
(0 - 4)^2 + (y - 9)^2 = 81
\]

This simplifies to:

\[
16 + (y - 9)^2 = 81
\]

\[
(y - 9)^2 = 81 - 16
\]

\[
(y - 9)^2 = 65
\]

\[
y - 9 = \pm \sqrt{65}
\]

\[
y = 9 \pm \sqrt{65}
\]

The two y-intercepts are:

\[
y = 9 + \sqrt{65} \quad \text{and} \quad y = 9 - \sqrt{65}
\]

Thus, the y-intercepts are \(\left(0, 9 + \sqrt{65}\right)\) and \(\left(0, 9 - \sqrt{65}\right)\).

### Summary:
- **X-intercept**: \((4, 0)\)
- **Y-intercepts**: \(\left(0, 9 + \sqrt{65}\right)\) and \(\left(0, 9 - \sqrt{65}\right)\)

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How do you find the radius of a circle from its equation?
2. What are the general steps to find x- and y-intercepts for other types of curves?
3. How can the equation of a circle be transformed or shifted?
4. What is the significance of the center of a circle in its equation?
5. How do you solve quadratic equations to find intercepts?

**Tip:** The intercepts of a curve provide critical points where the graph crosses the axes, giving insights into its behavior.

Find the​ x- and​ y- intercepts of the graph of the given equation: (x - 4)^2 + (y - 9)^2 = 81

Learn how to find the x- and y-intercepts of the circle equation (x - 4)^2 + (y - 9)^2 = 81. This problem explores intercepts, geometry, and algebraic solutions for high school students in Grades 9-11.

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