To determine whether the given point $(-4, 1)$ lies inside, outside, or on the circle defined by the equation $x^2 + y^2 + 2x + 8y - 13 = 0$, we follow these steps:

1. Rewrite the circle's equation in standard form:

The equation of the circle is given as: $x^2 + y^2 + 2x + 8y - 13 = 0$ We will complete the square for the $x$ and $y$ terms to rewrite this equation in the standard form $(x - h)^2 + (y - k)^2 = r^2$.

Complete the square for $x$:

$x^2 + 2x = (x + 1)^2 - 1$

Complete the square for $y$:

$y^2 + 8y = (y + 4)^2 - 16$

Substituting these into the original equation: $(x + 1)^2 - 1 + (y + 4)^2 - 16 - 13 = 0$ Simplifying: $(x + 1)^2 + (y + 4)^2 - 30 = 0$ $(x + 1)^2 + (y + 4)^2 = 30$

So, the standard form of the circle's equation is: $(x + 1)^2 + (y + 4)^2 = 30$ where the center of the circle $(h, k)$ is $(-1, -4)$ and the radius $r$ is $\sqrt{30}$.

2. Calculate the distance from the point $(-4, 1)$ to the center of the circle $(-1, -4)$:

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the coordinates: $d = \sqrt{((-4) - (-1))^2 + (1 - (-4))^2}$ $d = \sqrt{(-3)^2 + (5)^2}$ $d = \sqrt{9 + 25} = \sqrt{34}$

3. Compare the distance $d$ with the radius $r$:

• If $d < r$, the point is inside the circle.
• If $d = r$, the point is on the circle.
• If $d > r$, the point is outside the circle.

Here, $d = \sqrt{34}$ and $r = \sqrt{30}$.

Since $\sqrt{34} > \sqrt{30}$, the distance from the point to the center is greater than the radius, meaning the point $(-4, 1)$ lies outside the circle.

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Do you want more details or have any questions about this problem?

Here are 5 related questions to consider:

1. How would you find the distance between two points in 3D space?
2. What is the geometric interpretation of completing the square?
3. How can you determine the radius of a circle given an equation in general form?
4. Can you derive the equation of a circle given the center and a point on the circle?
5. How does changing the sign of the coefficients in the circle equation affect its graph?

Tip: When completing the square, always remember to balance the equation by adding or subtracting the same value on both sides.