The locus of the mid-points of the chords of the ellipse `x^2/a^2+y^2/b^2 =1` which pass through the positive end of major axis, 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 pass through the positive end of the major axis, we can follow these steps:

### Step 1: Understand the Ellipse
The given ellipse is represented by the equation:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a > b \). The positive end of the major axis is the point \( (a, 0) \).

### Step 2: Define the Midpoint of the Chord
Let the midpoint of the chord be \( (h, k) \).

### Step 3: Use the Chord Equation
The equation of the chord with midpoint \( (h, k) \) can be derived using the formula:
\[ \frac{hx}{a^2} + \frac{ky}{b^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} - 1 \]

### Step 4: Substitute the Point \( (a, 0) \)
Since the chord passes through the point \( (a, 0) \), we substitute \( x = a \) and \( y = 0 \) into the chord equation:
\[ \frac{ha}{a^2} + \frac{k \cdot 0}{b^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} - 1 \]
This simplifies to:
\[ \frac{h}{a} = \frac{h^2}{a^2} + \frac{k^2}{b^2} - 1 \]

### Step 5: Rearranging the Equation
We can rearrange the equation:
\[ \frac{h}{a} + 1 = \frac{h^2}{a^2} + \frac{k^2}{b^2} \]
Multiplying through by \( a^2 b^2 \) to eliminate the denominators gives:
\[ b^2 h + a^2 b^2 = h^2 b^2 + k^2 a^2 \]

### Step 6: Rearranging Further
Rearranging this equation leads to:
\[ h^2 b^2 - b^2 h + k^2 a^2 - a^2 b^2 = 0 \]

### Step 7: Completing the Square
To complete the square for the \( h \) terms:
\[ b^2(h^2 - h) + k^2 a^2 - a^2 b^2 = 0 \]
Completing the square for \( h \):
\[ b^2\left(h - \frac{1}{2}\right)^2 - \frac{b^2}{4} + k^2 a^2 - a^2 b^2 = 0 \]
This gives:
\[ b^2\left(h - \frac{1}{2}\right)^2 + k^2 a^2 = \frac{b^2}{4} + a^2 b^2 \]

### Step 8: Final Equation
The final equation can be expressed as:
\[ \frac{(h - \frac{a}{2})^2}{\frac{a^2}{4}} + \frac{k^2}{b^2} = 1 \]
This is the equation of an ellipse.

### Conclusion
Thus, the locus of the midpoints of the chords of the ellipse that pass through the positive end of the major axis is another ellipse.

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