To find the intercepts of the given equation $(x - 3)^2 + (y - 6)^2 = 58$, we'll solve for both the x-intercepts and y-intercepts separately.

Finding the x-intercepts:

For x-intercepts, we set $y = 0$ and solve for $x$.

Substitute $y = 0$ into the equation: $(x - 3)^2 + (0 - 6)^2 = 58$

Simplify: $(x - 3)^2 + 36 = 58$

Subtract 36 from both sides: $(x - 3)^2 = 22$

Take the square root of both sides: $x - 3 = \pm \sqrt{22}$

Solve for $x$: $x = 3 \pm \sqrt{22}$

Thus, the x-intercepts are: $x = 3 + \sqrt{22} \quad \text{and} \quad x = 3 - \sqrt{22}$

Finding the y-intercepts:

For y-intercepts, we set $x = 0$ and solve for $y$.

Substitute $x = 0$ into the equation: $(0 - 3)^2 + (y - 6)^2 = 58$

Simplify: $9 + (y - 6)^2 = 58$

Subtract 9 from both sides: $(y - 6)^2 = 49$

Take the square root of both sides: $y - 6 = \pm 7$

Solve for $y$: $y = 6 + 7 \quad \text{or} \quad y = 6 - 7$

Thus, the y-intercepts are: $y = 13 \quad \text{and} \quad y = -1$

Summary of Intercepts:

• x-intercepts: $x = 3 + \sqrt{22}$ and $x = 3 - \sqrt{22}$
• y-intercepts: $y = 13$ and $y = -1$

Would you like any additional details or have any questions?

Here are some related questions for further practice:

1. Find the intercepts of $(x + 4)^2 + (y - 5)^2 = 29$.
2. Determine the x-intercepts of $x^2 + y^2 = 25$.
3. Calculate the y-intercepts of $(x - 2)^2 + (y + 3)^2 = 50$.
4. Solve for the intercepts of the ellipse equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$.
5. Find the intercepts for the hyperbola $\frac{x^2}{16} - \frac{y^2}{25} = 1$.
6. Determine the intercepts of $(x - 1)^2 + (y + 2)^2 = 34$.
7. Calculate the x-intercepts of $x^2 + y^2 = 49$.
8. Find the intercepts of the circle equation $(x + 3)^2 + (y - 7)^2 = 65$.

Tip: When dealing with intercepts, always set the opposite variable to zero and solve the resulting equation carefully.