The focal distance of an end of the minor axis of the ellipse is k and the distance between the foci is 2h. Find the lengths of the semi-axes.

Answer

To solve the problem step by step, we will start by understanding the given information and then derive the lengths of the semi-axes of the ellipse.

### Step 1: Understand the given information
We are given:
- The focal distance of an end of the minor axis of the ellipse is \( k \).
- The distance between the foci is \( 2h \).

### Step 2: Define the ellipse parameters
For an ellipse centered at the origin, the semi-major axis is \( a \) and the semi-minor axis is \( b \). The foci of the ellipse are located at \( (ae, 0) \) and \( (-ae, 0) \), where \( e \) is the eccentricity of the ellipse.

### Step 3: Relate the distance between the foci to \( a \) and \( e \)
The distance between the foci is given by:
\[ 2ae = 2h \]
From this, we can simplify to find:
\[ ae = h \]

### Step 4: Use the focal distance of the minor axis
The focal distance of an end of the minor axis is given as \( k \). The coordinates of the end of the minor axis are \( (0, b) \) and \( (0, -b) \). The distance from this point to the focus is given by:
\[ \sqrt{(0 - ae)^2 + (b - 0)^2} = k \]
Squaring both sides gives:
\[ (ae)^2 + b^2 = k^2 \]

### Step 5: Substitute \( ae \) from Step 3
From Step 3, we know \( ae = h \). Therefore, we can substitute \( h \) into the equation:
\[ h^2 + b^2 = k^2 \]

### Step 6: Solve for \( b^2 \)
Rearranging the equation gives:
\[ b^2 = k^2 - h^2 \]

### Step 7: Use the relationship between \( a \), \( b \), and \( e \)
We also know that:
\[ b^2 = a^2(1 - e^2) \]
From this, we can express \( e^2 \) in terms of \( a \) and \( b \):
\[ 1 - e^2 = \frac{b^2}{a^2} \]
Substituting \( b^2 \) from Step 6 into this equation gives:
\[ 1 - e^2 = \frac{k^2 - h^2}{a^2} \]
This implies:
\[ e^2 = 1 - \frac{k^2 - h^2}{a^2} \]

### Step 8: Substitute \( e \) back into \( ae = h \)
From \( ae = h \), we can express \( e \) as:
\[ e = \frac{h}{a} \]
Substituting this into the equation for \( e^2 \) gives:
\[ \left(\frac{h}{a}\right)^2 = 1 - \frac{k^2 - h^2}{a^2} \]
This simplifies to:
\[ \frac{h^2}{a^2} = 1 - \frac{k^2 - h^2}{a^2} \]

### Step 9: Solve for \( a \)
Multiplying through by \( a^2 \) gives:
\[ h^2 = a^2 - (k^2 - h^2) \]
Rearranging gives:
\[ h^2 + k^2 = a^2 \]
Thus, we find:
\[ a = \sqrt{h^2 + k^2} \]

### Step 10: Find \( b \)
Using \( b^2 = k^2 - h^2 \):
\[ b = \sqrt{k^2 - h^2} \]

### Final Result
The lengths of the semi-axes are:
- Semi-major axis \( a = \sqrt{h^2 + k^2} \)
- Semi-minor axis \( b = \sqrt{k^2 - h^2} \)