If the focal distance of one ed of minor axis of an ellipse is k and distance betwnn foci is 2h then find the equation of the ellipse.

Answer

To find the equation of the ellipse given the focal distance of one end of the minor axis is \( k \) and the distance between the foci is \( 2h \), we can follow these steps:

### Step 1: Understand the parameters of the ellipse
The distance between the foci of the ellipse is given as \( 2h \). This means that the distance from the center to each focus (denoted as \( c \)) is:
\[ c = h \]

### Step 2: Relate the semi-major axis \( a \), semi-minor axis \( b \), and the focal distance \( c \)
For an ellipse, the relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the focal distance \( c \) is given by the equation:
\[ c^2 = a^2 - b^2 \]

### Step 3: Identify the semi-minor axis \( b \)
It is given that the minor axis of the ellipse is \( k \). Therefore, the semi-minor axis \( b \) is:
\[ b = \frac{k}{2} \]
Thus, we have:
\[ b^2 = \left(\frac{k}{2}\right)^2 = \frac{k^2}{4} \]

### Step 4: Substitute the values into the relationship
Now, substituting the values of \( c \) and \( b^2 \) into the equation \( c^2 = a^2 - b^2 \):
\[ h^2 = a^2 - \frac{k^2}{4} \]

### Step 5: Rearranging the equation to find \( a^2 \)
Rearranging the equation gives us:
\[ a^2 = h^2 + \frac{k^2}{4} \]

### Step 6: Write the equation of the ellipse
The standard form of the equation of an ellipse centered at the origin is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ \frac{x^2}{h^2 + \frac{k^2}{4}} + \frac{y^2}{\frac{k^2}{4}} = 1 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ \frac{x^2}{h^2 + \frac{k^2}{4}} + \frac{y^2}{\frac{k^2}{4}} = 1 \]