The equation of chord of the hyperbola `25x^2- 16y^2 =400` which is bisected at the point `(6, 2)` is

Options

A. `16x-75y=0`
B. `75x-16y=400`
C. `25x-4y=400`
D. None of these

Answer

To find the equation of the chord of the hyperbola \( 25x^2 - 16y^2 = 400 \) that is bisected at the point \( (6, 2) \), we can use the property of conics that relates the midpoint of a chord to the tangent at that point.

### Step-by-Step Solution:

1. **Identify the hyperbola equation**:
The given hyperbola is \( 25x^2 - 16y^2 = 400 \). We can rewrite it in standard form:
\[ \frac{x^2}{16} - \frac{y^2}{25} = 1 \]

2. **Find the coordinates of the midpoint**:
The midpoint of the chord is given as \( (6, 2) \).

3. **Use the formula for the chord bisected at a point**:
The equation of the chord of a conic that is bisected at the point \( (x_1, y_1) \) can be expressed as:
\[ T = S_1 \]
where \( T \) is the equation of the tangent at the point \( (x_1, y_1) \) and \( S_1 \) is the equation of the conic evaluated at \( (x_1, y_1) \).

4. **Calculate \( T \)**:
For the hyperbola \( 25x^2 - 16y^2 = 400 \), the tangent at the point \( (x_1, y_1) \) is given by:
\[ T: 25xx_1 - 16yy_1 = 400 \]
Substituting \( (x_1, y_1) = (6, 2) \):
\[ T: 25x \cdot 6 - 16y \cdot 2 = 400 \]
Simplifying this:
\[ 150x - 32y = 400 \]

5. **Calculate \( S_1 \)**:
Now, we evaluate \( S_1 \) at the point \( (6, 2) \):
\[ S_1: 25(6^2) - 16(2^2) - 400 = 0 \]
Calculating this:
\[ S_1: 25 \cdot 36 - 16 \cdot 4 - 400 = 900 - 64 - 400 = 436 \]

6. **Set up the equation**:
Now, we set \( T = S_1 \):
\[ 150x - 32y = 436 \]

7. **Final equation**:
Rearranging gives us the equation of the chord:
\[ 150x - 32y - 436 = 0 \]

### Final Answer:
The equation of the chord of the hyperbola that is bisected at the point \( (6, 2) \) is:
\[ 150x - 32y - 436 = 0 \]