If the line pass through the point(-3,-5) and intersect the ellipse `x^2/4+y^2/9=1` at a point A and B . Then the locus of mid point of line joining A and B is

Options

A. `4x^2+9y^2+20x+27y=0`
B. `9x^2+4y^2+20x+27y=0`
C. `4x^2+9y^2+27x+20y=0`
D. `9x^2+4y^2+27x+20y=0`

Answer

To find the locus of the midpoint of the line segment joining points A and B, where the line passes through the point (-3, -5) and intersects the ellipse given by the equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \), we can follow these steps:

### Step 1: Equation of the Ellipse
The equation of the ellipse is given as:
\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]
This represents an ellipse centered at the origin with semi-major axis 3 (along the y-axis) and semi-minor axis 2 (along the x-axis).

### Step 2: Equation of the Chord
The chord of the ellipse that passes through the point (-3, -5) can be expressed using the chord equation:
\[ \frac{hx}{4} + \frac{ky}{9} = h^2/4 + k^2/9 - 27h - 20k \]
where (h, k) is the midpoint of the chord AB.

### Step 3: Substitute the Point (-3, -5)
Since the line passes through the point (-3, -5), we substitute \( x = -3 \) and \( y = -5 \) into the chord equation:
\[ \frac{h(-3)}{4} + \frac{k(-5)}{9} = \frac{h^2}{4} + \frac{k^2}{9} - 27h - 20k \]

### Step 4: Simplifying the Equation
This gives us:
\[ -\frac{3h}{4} - \frac{5k}{9} = \frac{h^2}{4} + \frac{k^2}{9} - 27h - 20k \]
Multiplying through by 36 (the least common multiple of 4 and 9) to eliminate the denominators:
\[ -27h - 20k = 9h^2 + 4k^2 - 972h - 720k \]

### Step 5: Rearranging the Equation
Rearranging the equation leads to:
\[ 9h^2 + 4k^2 + 945h + 700k = 0 \]

### Step 6: Locus Equation
This is the equation of the locus of the midpoint (h, k) of the line segment AB.

### Final Locus Equation
The final equation can be expressed as:
\[ 9h^2 + 4k^2 + 945h + 700k = 0 \]