Find the locus of the middle points of chords of an ellipse the tangents at the ends of which intersect at right angles.

Answer

To find the locus of the midpoints of chords of an ellipse where the tangents at the ends of the chords intersect at right angles, we can follow these steps:

### Step 1: Define the Ellipse
We start with the standard equation of the ellipse:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.

### Step 2: Midpoint of the Chord
Let \(P(h, k)\) be the midpoint of the chord \(AB\) on the ellipse. The equation of the chord can be expressed using the midpoint formula:
\[ T = S_1 \]
where \(T\) is the equation of the chord and \(S_1\) is the equation of the ellipse evaluated at the midpoint.

### Step 3: Write the Equation of the Chord
Using the midpoint \(P(h, k)\), we can write the equation of the chord as:
\[ \frac{hx}{a^2} + \frac{ky}{b^2} = 1 \]

### Step 4: Tangents at the Ends of the Chord
Let the points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) be the endpoints of the chord. The tangents at these points can be represented as:
\[ \frac{x_1 x}{a^2} + \frac{y_1 y}{b^2} = 1 \quad \text{and} \quad \frac{x_2 x}{a^2} + \frac{y_2 y}{b^2} = 1 \]

### Step 5: Condition for Perpendicular Tangents
For the tangents at points \(A\) and \(B\) to intersect at right angles, the product of their slopes must equal \(-1\). This leads to the condition:
\[ \frac{y_1}{b^2} \cdot \frac{y_2}{b^2} + \frac{x_1}{a^2} \cdot \frac{x_2}{a^2} = 0 \]

### Step 6: Substitute the Coordinates
Using the ellipse equation, we can express \(y_1\) and \(y_2\) in terms of \(x_1\) and \(x_2\):
\[ y_1 = b\sqrt{1 - \frac{x_1^2}{a^2}}, \quad y_2 = b\sqrt{1 - \frac{x_2^2}{a^2}} \]

### Step 7: Substitute and Simplify
Substituting these expressions into the perpendicular condition and simplifying, we can derive a relationship involving \(h\) and \(k\).

### Step 8: Final Equation
After simplification, we will arrive at the equation:
\[ \frac{h^2}{a^2} + \frac{k^2}{b^2} = 1 \]
This represents an ellipse, which is the locus of the midpoints of the chords.

### Conclusion
Thus, the locus of the midpoints of the chords of the ellipse, where the tangents at the ends of the chords intersect at right angles, is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]