Find the locus of the middle points of chords of an ellipse which subtend a right angle at the centre

Answer

To find the locus of the midpoints of the chords of an ellipse that subtend a right angle at the center, we can follow these steps:

### Step 1: Write the equation of the ellipse
The standard form of the equation of an ellipse centered at the origin is given by:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.

### Step 2: Define the midpoint of the chord
Let \(P(\alpha, \beta)\) be the midpoint of a chord \(AB\) of the ellipse. The coordinates of the midpoint \(P\) can be expressed as:

\[ P\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

where \((x_1, y_1)\) and \((x_2, y_2)\) are the endpoints of the chord.

### Step 3: Use the property of the chord subtending a right angle
Since the chord \(AB\) subtends a right angle at the center (the origin), we can use the property that the product of the slopes of the lines connecting the origin to the endpoints of the chord is \(-1\):

\[ \frac{y_1}{x_1} \cdot \frac{y_2}{x_2} = -1 \]

This implies:

\[ y_1 y_2 = -x_1 x_2 \]

### Step 4: Express \(x_1\) and \(y_1\) in terms of \(\alpha\) and \(\beta\)
From the midpoint formula, we can express \(x_1\) and \(y_1\) in terms of \(\alpha\) and \(\beta\):

\[ x_1 = 2\alpha - x_2 \]
\[ y_1 = 2\beta - y_2 \]

### Step 5: Substitute into the ellipse equation
Substituting \(x_1\) and \(y_1\) into the ellipse equation gives:

\[ \frac{(2\alpha - x_2)^2}{a^2} + \frac{(2\beta - y_2)^2}{b^2} = 1 \]

### Step 6: Homogenize and simplify
To find the locus, we need to eliminate \(x_2\) and \(y_2\) using the relationship derived from the right angle condition:

Substituting \(y_2 = -\frac{y_1}{x_1}x_2\) into the ellipse equation and simplifying will yield a relationship involving \(\alpha\) and \(\beta\).

### Step 7: Find the locus equation
After simplification, we arrive at the locus equation:

\[ \frac{\alpha^2}{\frac{a^4}{b^2 + a^2}} + \frac{\beta^2}{\frac{b^4}{a^2 + b^2}} = 1 \]

### Final Step: Replace \(\alpha\) and \(\beta\)
Finally, replace \(\alpha\) and \(\beta\) with \(x\) and \(y\) to express the locus in standard form:

\[ \frac{x^2}{\frac{a^4}{b^2 + a^2}} + \frac{y^2}{\frac{b^4}{a^2 + b^2}} = 1 \]

This is the required locus of the midpoints of the chords of the ellipse that subtend a right angle at the center.