The locus of mid-points of a focal chord of the ellipse `x^2/a^2+y^2/b^2=1`

Options

A. `(x^2)/(a^2)+(y^2)/(b^2)=(ex)/(a)`
B. `(x^2)/(a^2)-(y^2)/(b^2)=(ex)/(a)`
C. `x^2+y^2=a^2+b^2`
D. `(x^2)/(a^2)+(y^2)/(b^2)=(ex)/(a)`

Answer

To find the locus of midpoints of a focal chord of the ellipse given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we can follow these steps:

### Step 1: Understand the ellipse and its foci
The given ellipse has the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). The foci of the ellipse are located at \( (c, 0) \) and \( (-c, 0) \), where \( c = \sqrt{a^2 - b^2} \).

### Step 2: Define the focal chord
A focal chord is a line segment that passes through one of the foci of the ellipse and has its endpoints on the ellipse. Let's denote the endpoints of the focal chord as \( P(t_1) \) and \( P(t_2) \), where \( t_1 \) and \( t_2 \) are parameters corresponding to the points on the ellipse.

### Step 3: Parametric equations of the ellipse
The parametric equations for the ellipse are:
\[ x = a \cos t, \quad y = b \sin t \]
Thus, the coordinates of the endpoints of the focal chord can be expressed as:
\[ P(t_1) = (a \cos t_1, b \sin t_1) \quad \text{and} \quad P(t_2) = (a \cos t_2, b \sin t_2) \]

### Step 4: Midpoint of the focal chord
The midpoint \( M \) of the focal chord \( P(t_1) \) and \( P(t_2) \) is given by:
\[ M = \left( \frac{a \cos t_1 + a \cos t_2}{2}, \frac{b \sin t_1 + b \sin t_2}{2} \right) \]

### Step 5: Use the property of focal chords
For a focal chord, it is known that if \( t_1 \) and \( t_2 \) are the parameters of the endpoints, then \( t_2 = -t_1 \). Therefore, we can substitute \( t_2 = -t_1 \):
\[ M = \left( \frac{a \cos t_1 + a \cos(-t_1)}{2}, \frac{b \sin t_1 + b \sin(-t_1)}{2} \right) \]
Using the even and odd properties of cosine and sine:
\[ M = \left( \frac{a \cos t_1 + a \cos t_1}{2}, \frac{b \sin t_1 - b \sin t_1}{2} \right) = \left( a \cos t_1, 0 \right) \]

### Step 6: Locus of the midpoints
The x-coordinate of the midpoint is \( h = a \cos t_1 \) and the y-coordinate is \( k = 0 \). Therefore, the locus of the midpoints of the focal chords is along the x-axis, given by:
\[ k = 0 \]
This means that the locus of midpoints of the focal chords is the line \( y = 0 \).

### Final Result
Thus, the locus of midpoints of the focal chords of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is:
\[ y = 0 \]