Find in each case whether the given point lies inside, outside or on the given circle. (−4, 1) , x^2 + y^2 + 2x + 8y − 13 = 0

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.

Discover how to determine if a given point lies inside, outside, or on a circle using an analytical approach. Solve the problem step-by-step with the circle equation x^2 + y^2 + 2x + 8y - 13 = 0 and point (-4, 1). Explore completing the square, distance calculations, and geometric interpretations.

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