https://file.aiquickdraw.com/mathbot/file/1726464597238owsbksz6.jpg

Let's work through the three visible problems on the image.

### Problem 53: Weather Satellites
The Earth is represented by the equation of a circle:
\[
x^2 + y^2 + 10x + 8y - 6359 = 0
\]
A weather satellite orbits at 0.7 units above Earth. We need to find the general equation for the satellite's orbit.

#### Step 1: Complete the square to rewrite Earth's equation in standard form.
The equation of the Earth is:
\[
x^2 + y^2 + 10x + 8y - 6359 = 0
\]
Group the terms by \(x\) and \(y\):
\[
(x^2 + 10x) + (y^2 + 8y) = 6359
\]
Complete the square for both \(x\) and \(y\):
- For \(x\): \((x^2 + 10x) \Rightarrow (x + 5)^2 - 25\)
- For \(y\): \((y^2 + 8y) \Rightarrow (y + 4)^2 - 16\)

Thus, the equation becomes:
\[
(x + 5)^2 - 25 + (y + 4)^2 - 16 = 6359
\]
Simplifying:
\[
(x + 5)^2 + (y + 4)^2 = 6400
\]
This represents a circle with center \((-5, -4)\) and radius \(80\) units.

#### Step 2: Adjust for the satellite's orbit.
The satellite orbits at 0.7 units above the surface. Therefore, the radius of the satellite's orbit is:
\[
80 + 0.7 = 80.7 \, \text{units}
\]
Thus, the equation for the satellite's orbit is:
\[
(x + 5)^2 + (y + 4)^2 = 80.7^2
\]

### Problem 58: Equation of the Line Connecting Two Circles
We are given two circle equations:
1. \(x^2 + y^2 - 4x + 6y + 4 = 0\)
2. \(x^2 + y^2 + 6x + 4y + 9 = 0\)

#### Step 1: Convert both to standard form by completing the square.
For the first circle:
\[
x^2 - 4x + y^2 + 6y = -4
\]
Complete the square for \(x\) and \(y\):
- \(x^2 - 4x = (x - 2)^2 - 4\)
- \(y^2 + 6y = (y + 3)^2 - 9\)

Thus, the first equation becomes:
\[
(x - 2)^2 + (y + 3)^2 = 9
\]
This represents a circle with center \((2, -3)\) and radius \(3\).

For the second circle:
\[
x^2 + 6x + y^2 + 4y = -9
\]
Complete the square for \(x\) and \(y\):
- \(x^2 + 6x = (x + 3)^2 - 9\)
- \(y^2 + 4y = (y + 2)^2 - 4\)

Thus, the second equation becomes:
\[
(x + 3)^2 + (y + 2)^2 = 4
\]
This represents a circle with center \((-3, -2)\) and radius \(2\).

#### Step 2: Find the equation of the line connecting the centers.
The centers of the circles are \((2, -3)\) and \((-3, -2)\). The slope \(m\) of the line connecting these points is:
\[
m = \frac{-2 - (-3)}{-3 - 2} = \frac{1}{-5} = -\frac{1}{5}
\]
Using the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
Substitute \((2, -3)\) and \(m = -\frac{1}{5}\):
\[
y + 3 = -\frac{1}{5}(x - 2)
\]
Simplifying:
\[
y + 3 = -\frac{1}{5}x + \frac{2}{5}
\]
Multiply by 5 to clear the fraction:
\[
5y + 15 = -x + 2
\]
Rearranging:
\[
x + 5y = -13
\]
Thus, the equation of the line is:
\[
x + 5y = -13
\]

Would you like to go over Problem 51 in more detail? Let me know if you want further clarification!

Here are some related questions:
1. How do you complete the square for equations of circles?
2. What is the geometric significance of a line connecting the centers of two circles?
3. How does a change in radius affect the equation of a circle?
4. What is the standard form of the equation of a circle?
5. How can you find the distance between two points in a coordinate plane?

**Tip:** When dealing with geometric figures like circles, always aim to simplify the equation into a recognizable form by completing the square or using standard formulas.

53. A weather satellite orbits 0.7 units above the Earth, represented by the equation x^2 + y^2 + 10x + 8y - 6359 = 0. Find the general form of the equation for the orbit of the satellite.

Find the equation for the orbit of a weather satellite circling 0.7 units above the Earth. This problem involves completing the square to rewrite the equation of the Earth, calculating the satellite's orbit, and finding the general form of a circle equation. Suitable for students in Grades 10-12.

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