The locus of extremities of the latus rectum of the family of ellipse `b^2x^2+a^2y^2=a^2b^2` is

Answer

To find the locus of the extremities of the latus rectum of the family of ellipses given by the equation \( b^2x^2 + a^2y^2 = a^2b^2 \), we will follow these steps:

### Step 1: Rewrite the equation of the ellipse
The given equation of the ellipse is:
\[ b^2x^2 + a^2y^2 = a^2b^2 \]
We can divide the entire equation by \( a^2 \) to simplify it:
\[ \frac{b^2}{a^2}x^2 + y^2 = b^2 \]
This can be rewritten as:
\[ \frac{x^2}{\frac{a^2}{b^2}} + \frac{y^2}{b^2} = 1 \]
This represents an ellipse centered at the origin with semi-major axis \( b \) and semi-minor axis \( \frac{a}{b} \).

### Step 2: Identify the coordinates of the extremities of the latus rectum
The latus rectum of an ellipse is a line segment perpendicular to the major axis that passes through a focus. The coordinates of the extremities of the latus rectum for an ellipse can be given by:
\[ \left( ae, \frac{b^2}{a} \right) \quad \text{and} \quad \left( ae, -\frac{b^2}{a} \right) \]
where \( e \) is the eccentricity of the ellipse, defined as:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

### Step 3: Substitute the value of eccentricity
Substituting \( e \) into the coordinates:
\[ H = ae = a\sqrt{1 - \frac{b^2}{a^2}} = a\sqrt{\frac{a^2 - b^2}{a^2}} = \sqrt{a^2 - b^2} \]
Thus, the coordinates of the extremities of the latus rectum become:
\[ \left( \sqrt{a^2 - b^2}, \frac{b^2}{a} \right) \quad \text{and} \quad \left( \sqrt{a^2 - b^2}, -\frac{b^2}{a} \right) \]

### Step 4: Find the locus
Let \( H = \sqrt{a^2 - b^2} \) and \( K = \frac{b^2}{a} \). We can express \( b^2 \) in terms of \( H \):
\[ b^2 = a^2 - H^2 \]
Substituting this into the expression for \( K \):
\[ K = \frac{a^2 - H^2}{a} \]
This gives us:
\[ K = a - \frac{H^2}{a} \]
Rearranging gives:
\[ H^2 + aK - a^2 = 0 \]

### Step 5: Final equation of the locus
Replacing \( H \) with \( x \) and \( K \) with \( y \):
\[ x^2 + ay - a^2 = 0 \]
This is the equation of the locus of the extremities of the latus rectum.