The locus of he middle points of the chords of the ellipse `x^(2)/a^(2)+y^(2)/b^(2)=1` which are at a constant disance 'd' form the cantre is

Options

A. `(x^(2)/a^(2)+y^(2)/b^(2))^(2)=d^(2)(x^(2)/a^(2)-y^(2)/b^(2))`
B. `(x^(2)/a^(2)+y^(2)/b^(2))^(2)=d^(2)(x^(2)/a^(4)+y^(2)/b^(4))`
C. `(x^(2)/a^(2)-y^(2)/b^(2))^(2)=d^(2)(x^(2)/a^(4)+y^(2)/b^(4))`
D. None of these

Answer

To find the locus of the midpoints of the chords of the ellipse given by the equation

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

which are at a constant distance \(d\) from the center (0, 0), we can follow these steps:

### Step 1: Understand the midpoint of a chord
Let the midpoint of the chord be \(M(x_1, y_1)\). The equation of the chord of the ellipse with midpoint \(M\) is given by:

\[ T = S_1 \]

where \(T\) is the equation of the chord and \(S_1\) is the value of the ellipse at the midpoint.

### Step 2: Write the equation of the chord
The equation of the chord can be expressed as:

\[ \frac{x \cdot x_1}{a^2} + \frac{y \cdot y_1}{b^2} = \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} \]

### Step 3: Find the distance from the center
The distance from the center (0, 0) to the point \(M(x_1, y_1)\) is given by:

\[ d = \sqrt{x_1^2 + y_1^2} \]

### Step 4: Set up the condition for the distance
Since the distance \(d\) is constant, we can write:

\[ \sqrt{x_1^2 + y_1^2} = d \]

Squaring both sides gives:

\[ x_1^2 + y_1^2 = d^2 \]

### Step 5: Use the ellipse equation
From the ellipse equation, we can express the relationship between \(x_1\) and \(y_1\):

\[ \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1 \]

### Step 6: Substitute \(y_1^2\) in terms of \(x_1^2\)
From the distance equation, we can express \(y_1^2\) as:

\[ y_1^2 = d^2 - x_1^2 \]

### Step 7: Substitute into the ellipse equation
Substituting \(y_1^2\) into the ellipse equation gives:

\[ \frac{x_1^2}{a^2} + \frac{d^2 - x_1^2}{b^2} = 1 \]

### Step 8: Rearranging the equation
Rearranging this equation leads to:

\[ \frac{x_1^2}{a^2} + \frac{d^2}{b^2} - \frac{x_1^2}{b^2} = 1 \]

### Step 9: Combine like terms
Combining the terms involving \(x_1^2\):

\[ x_1^2 \left(\frac{1}{a^2} - \frac{1}{b^2}\right) + \frac{d^2}{b^2} = 1 \]

### Step 10: Solve for \(x_1^2\)
This can be rearranged to express \(x_1^2\):

\[ x_1^2 = \frac{(1 - \frac{d^2}{b^2})}{(\frac{1}{a^2} - \frac{1}{b^2})} \]

### Step 11: Find the locus
The locus of the midpoints \(M(x_1, y_1)\) can be derived from the above equation, leading to the final equation of the locus.

### Final Result
The locus of the midpoints of the chords of the ellipse at a constant distance \(d\) from the center is:

\[ \frac{x^2}{\left(\frac{d^2}{b^2}\right)} + \frac{y^2}{\left(\frac{d^2}{a^2}\right)} = 1 \]