To find the foci of the ellipse given the equations of the major and minor axes and one end of the latus rectum, we can follow these steps:
### Step 1: Identify the equations of the axes
The equations of the major axis and minor axis are given as:
- Major axis: \( x - 2y - 5 = 0 \)
- Minor axis: \( 2x + y + 10 = 0 \)
### Step 2: Find the intersection point of the axes
To find the center of the ellipse, we need to solve these two equations simultaneously.
1. From the first equation, express \( x \) in terms of \( y \):
\[
x = 2y + 5
\]
2. Substitute \( x \) into the second equation:
\[
2(2y + 5) + y + 10 = 0
\]
\[
4y + 10 + y + 10 = 0
\]
\[
5y + 20 = 0 \implies y = -4
\]
3. Substitute \( y = -4 \) back into the expression for \( x \):
\[
x = 2(-4) + 5 = -8 + 5 = -3
\]
Thus, the intersection point (center of the ellipse) is:
\[
(-3, -4)
\]
### Step 3: Find the coordinates of the focus using the end of the latus rectum
Given one end of the latus rectum is \( (3, 4) \), we can denote the focus as \( (h, k) \).
The latus rectum is perpendicular to the major axis, so we can use the property of the ellipse that states the distance from the center to the focus is equal to \( c \), where \( c^2 = a^2 - b^2 \).
### Step 4: Set up the equations based on the latus rectum
The coordinates of the focus \( (h, k) \) can be expressed in terms of the coordinates of the end of the latus rectum:
- The midpoint of the latus rectum is the focus. Therefore, if one end is \( (3, 4) \), the other end can be denoted as \( (x', y') \).
Using the midpoint formula:
\[
\left( \frac{3 + x'}{2}, \frac{4 + y'}{2} \right) = (h, k)
\]
### Step 5: Use the equations of the axes to find the slope
The slope of the major axis can be calculated from its equation:
- Rearranging \( x - 2y - 5 = 0 \) gives \( y = \frac{1}{2}x - \frac{5}{2} \) (slope = \( \frac{1}{2} \)).
- The slope of the minor axis is from \( 2x + y + 10 = 0 \), which gives \( y = -2x - 10 \) (slope = \( -2 \)).
### Step 6: Solve for the focus coordinates
Using the perpendicularity of the latus rectum to the major axis, we can find the coordinates of the focus. We know:
\[
h - 3 = m (k - 4)
\]
Where \( m = -2 \) (the negative reciprocal of the slope of the major axis).
Substituting and solving gives:
1. \( h - 3 = -2(k - 4) \)
2. Rearranging gives \( h = -2k + 8 + 3 = -2k + 11 \).
### Step 7: Use the midpoint to find the second focus
From the midpoint equation:
\[
(-3, -4) = \left( \frac{3 + x'}{2}, \frac{4 + y'}{2} \right)
\]
This gives us two equations:
1. \( -3 = \frac{3 + x'}{2} \) leading to \( x' = -9 \).
2. \( -4 = \frac{4 + y'}{2} \) leading to \( y' = -12 \).
### Step 8: Find the coordinates of both foci
Now we have:
- Focus 1 \( (h, k) = (5, 0) \)
- Focus 2 \( (x', y') = (-11, -8) \)
### Final Answer
The foci of the ellipse are:
\[
F_1 = (5, 0) \quad \text{and} \quad F_2 = (-11, -8)
\]
