Find the locus of the middle points of chords of an ellipse whose distance from the centre is the constant length c

Answer

To find the locus of the midpoints of chords of an ellipse whose distance from the center is a constant length \( c \), we can follow these steps:

### Step-by-Step Solution:

1. **Equation of the Ellipse**:
The standard form of the 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.

2. **Midpoint of the Chord**:
Let \( P(h, k) \) be the midpoint of a chord \( AB \) of the ellipse. The distance from the center (origin) to the point \( P \) is given to be a constant \( c \). Thus, we have:
\[ \sqrt{h^2 + k^2} = c \]
Squaring both sides, we get:
\[ h^2 + k^2 = c^2 \]

3. **Chord Equation**:
The equation of the chord with midpoint \( P(h, k) \) can be expressed using the formula:
\[ \frac{xh}{a^2} + \frac{yk}{b^2} = 1 \]

4. **Using the Condition of the Chord**:
We will apply the condition that this chord must intersect the ellipse. The condition for a point \( (x, y) \) to lie on the ellipse is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
For the midpoint \( P(h, k) \), we can use the relation:
\[ \frac{h^2}{a^2} + \frac{k^2}{b^2} = 1 \]

5. **Substituting \( h^2 \) and \( k^2 \)**:
From the previous steps, we have:
\[ h^2 = c^2 - k^2 \]
Substitute \( h^2 \) in the chord condition:
\[ \frac{c^2 - k^2}{a^2} + \frac{k^2}{b^2} = 1 \]

6. **Rearranging the Equation**:
Multiply through by \( a^2b^2 \) to eliminate the denominators:
\[ b^2(c^2 - k^2) + a^2k^2 = a^2b^2 \]
Simplifying gives:
\[ b^2c^2 - b^2k^2 + a^2k^2 = a^2b^2 \]
\[ b^2c^2 = a^2b^2 + (a^2 - b^2)k^2 \]

7. **Solving for \( k^2 \)**:
Rearranging gives:
\[ (a^2 - b^2)k^2 = b^2c^2 - a^2b^2 \]
\[ k^2 = \frac{b^2(c^2 - a^2)}{a^2 - b^2} \]

8. **Finding the Locus**:
Substitute \( k^2 \) back into the equation \( h^2 + k^2 = c^2 \):
\[ h^2 + \frac{b^2(c^2 - a^2)}{a^2 - b^2} = c^2 \]
Rearranging this gives the locus of the midpoints of the chords.

9. **Final Equation**:
The final locus equation can be expressed in terms of \( h \) and \( k \) (or \( x \) and \( y \)):
\[ \frac{h^2}{\frac{c^2b^2}{a^2 - b^2}} + \frac{k^2}{\frac{c^2a^2}{b^2 - a^2}} = 1 \]