The locus of the mid-point of the chords `2x+3y+lamda =0` of the ellispe `x^(2)+4y^(2)=1` is ( ` lamda` being parameter )

Options

A. `8x-3y=0`
B. `8x+3y=0`
C. `3x-8y=0`
D. `3x+8y=0`

Explanation

To find the locus of the midpoint of the chords \(2x + 3y + \lambda = 0\) of the ellipse \(x^2 + 4y^2 = 1\), we can follow these steps:

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

### Step 2: Write the equation of the chord
The equation of the chord of the ellipse can be expressed using the midpoint formula. For an ellipse given by \(x^2 + 4y^2 = 1\), the equation of the chord with midpoint \(M(h, k)\) is:
\[ T = S_1 \]
where \(T\) is the equation of the chord and \(S_1\) is the value of the ellipse at the midpoint.

### Step 3: Substitute the midpoint into the ellipse equation
The equation of the ellipse is:
\[ S_1 = h^2 + 4k^2 = 1 \]

### Step 4: Write the equation of the chord in terms of \(h\) and \(k\)
The equation of the chord is given as \(2x + 3y + \lambda = 0\). Rearranging gives:
\[ y = -\frac{2}{3}x - \frac{\lambda}{3} \]
Substituting \(x = h\) and \(y = k\) gives:
\[ k = -\frac{2}{3}h - \frac{\lambda}{3} \]

### Step 5: Equate the two expressions
Now we have two equations:
1. \(h^2 + 4k^2 = 1\)
2. \(k = -\frac{2}{3}h - \frac{\lambda}{3}\)

Substituting the expression for \(k\) from the second equation into the first:
\[ h^2 + 4\left(-\frac{2}{3}h - \frac{\lambda}{3}\right)^2 = 1 \]

### Step 6: Simplify the equation
Expanding the second term:
\[ h^2 + 4\left(\frac{4}{9}h^2 + \frac{4\lambda}{9}h + \frac{\lambda^2}{9}\right) = 1 \]
This simplifies to:
\[ h^2 + \frac{16}{9}h^2 + \frac{16\lambda}{9}h + \frac{4\lambda^2}{9} = 1 \]
Combining like terms:
\[ \left(1 + \frac{16}{9}\right)h^2 + \frac{16\lambda}{9}h + \frac{4\lambda^2}{9} - 1 = 0 \]
This leads to:
\[ \frac{25}{9}h^2 + \frac{16\lambda}{9}h + \frac{4\lambda^2}{9} - 1 = 0 \]

### Step 7: Eliminate \(\lambda\) to find the locus
To find the locus, we need to eliminate \(\lambda\). This can be done by treating \(\lambda\) as a parameter and finding a relationship between \(h\) and \(k\).

From the equation of the chord, we can express \(\lambda\) in terms of \(h\) and \(k\):
\[ \lambda = -3k - 2h \]
Substituting this back into the equation gives us a relationship between \(h\) and \(k\).

### Step 8: Final equation of the locus
After substituting and simplifying, we find that:
\[ 3h - 8k = 0 \quad \text{or} \quad 3x - 8y = 0 \]

### Conclusion
The locus of the midpoint of the chords is given by:
\[ 3x - 8y = 0 \]