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

Options

A. `16x-75y=418`
B. `75x-16y=418`
C. `25x-4y=400`
D. none of these

Answer

To find the equation of the chord of the hyperbola \( 25x^2 - 16y^2 = 400 \) which is bisected at the point \( (6, 2) \), we can follow these steps:

### Step 1: Rewrite the hyperbola in standard form
The given hyperbola equation is:
\[ 25x^2 - 16y^2 = 400 \]
Dividing the entire equation by 400 gives:
\[ \frac{x^2}{16} - \frac{y^2}{25} = 1 \]
This is the standard form of the hyperbola.

### Step 2: Identify the coordinates of the midpoint
The midpoint of the chord is given as \( (6, 2) \). We will denote this point as \( (x_1, y_1) \) where \( x_1 = 6 \) and \( y_1 = 2 \).

### Step 3: Use the formula for the equation of the chord
The equation of the chord of the hyperbola that is bisected at the point \( (x_1, y_1) \) can be expressed using the formula:
\[ T = S_1 \]
where \( T \) is the equation of the chord and \( S_1 \) is the value of the hyperbola at the midpoint.

### Step 4: Find \( S_1 \)
To find \( S_1 \), substitute \( (x_1, y_1) = (6, 2) \) into the hyperbola equation:
\[ S_1 = \frac{25(6)^2}{400} - \frac{16(2)^2}{400} \]
Calculating this gives:
\[ S_1 = \frac{25 \cdot 36}{400} - \frac{16 \cdot 4}{400} \]
\[ S_1 = \frac{900}{400} - \frac{64}{400} \]
\[ S_1 = \frac{900 - 64}{400} = \frac{836}{400} = \frac{209}{100} \]

### Step 5: Set up the equation of the chord
Using the formula \( T = S_1 \), we have:
\[ T = \frac{25x^2}{400} - \frac{16y^2}{400} = \frac{209}{100} \]
Multiplying through by 400 gives:
\[ 25x^2 - 16y^2 = 836 \]

### Step 6: Final equation of the chord
Thus, the equation of the chord of the hyperbola that is bisected at the point \( (6, 2) \) is:
\[ 25x^2 - 16y^2 = 836 \]