The locus of the middle points of the chords of the ellipse `x^2//a^2 + y^2//b^2 =1` touching the ellipse `x^2//alpha^2 + y^2//beta^2 = 1` is....... .

Answer

To find the locus of the midpoints of the chords of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) that touch the ellipse \( \frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} = 1 \), we can follow these steps:

### Step 1: Define the midpoint
Let the coordinates of the midpoint of the chord be \( (h, k) \).

### Step 2: Write the equation of the chord
The equation of the chord of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) can be expressed using the midpoint formula:
\[ \frac{xh}{a^2} + \frac{yk}{b^2} = 1 \]
This can be rearranged to:
\[ \frac{yk}{b^2} = 1 - \frac{xh}{a^2} \]

### Step 3: Substitute the midpoint into the equation of the ellipse
Now, substituting \( (h, k) \) into the equation of the ellipse, we have:
\[ \frac{h^2}{a^2} + \frac{k^2}{b^2} = 1 \]

### Step 4: Condition for tangency
For the chord to touch the ellipse \( \frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} = 1 \), we can use the condition of tangency. The distance from the center to the line must equal the semi-minor axis of the ellipse. The equation of the line can be written as:
\[ \frac{yk}{b^2} = 1 - \frac{xh}{a^2} \]
Rearranging gives:
\[ yk = b^2 - \frac{b^2xh}{a^2} \]

### Step 5: Find the relationship between the parameters
The condition for the line to be tangent to the ellipse \( \frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} = 1 \) can be expressed as:
\[ \left( \frac{b^2}{k} \right)^2 = \frac{b^4h^2}{a^4} + \frac{b^4}{\beta^2} \]

### Step 6: Simplify the equation
After substituting and simplifying, we arrive at the equation:
\[ \frac{h^2}{\alpha^2} + \frac{k^2}{\beta^2} = 1 \]

### Final Result
Thus, the locus of the midpoints of the chords of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) that touch the ellipse \( \frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} = 1 \) is given by:
\[ \frac{h^2}{\alpha^2} + \frac{k^2}{\beta^2} = 1 \]