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.