Let `H : x^(2) - y^(2) = 9, P : y^(2) = 4(x - 5), L : x = 9` be three curves. If L is the chord of contact of the hyperbola H, then the equation of the corresponding pair of tangent is

Options

A. (A)`9x^(2) - 8y^(2) + 18x + 9 = 0`
B. (B)`9x^(2) - 8y^(2) - 18x - 9 = 0`
C. (C)`9x^(2) - 8y^(2) - 18x + 9 = 0`
D. (D)`9x^(2) + 8y^(2) + 18x + 9 = 0`

Answer

To find the equation of the corresponding pair of tangents to the hyperbola \( H: x^2 - y^2 = 9 \) that correspond to the line \( L: x = 9 \), we will follow these steps:

### Step 1: Identify the hyperbola and its parameters
The given hyperbola is \( H: x^2 - y^2 = 9 \). We can rewrite this in standard form:
\[ \frac{x^2}{9} - \frac{y^2}{9} = 1 \]
From this, we can see that \( a^2 = 9 \) and \( b^2 = 9 \), hence \( a = 3 \) and \( b = 3 \).

### Step 2: Determine the point of contact
The line \( L: x = 9 \) is vertical. The point of contact on the hyperbola can be represented as \( (h, k) \). Since the line is vertical, we can take \( h = 9 \).

### Step 3: Use the formula for the chord of contact
The equation of the chord of contact from a point \( (h, k) \) to the hyperbola \( H \) is given by:
\[ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 \]
Substituting \( (h, k) = (9, k) \) into the equation:
\[ \frac{x \cdot 9}{9} - \frac{y \cdot k}{9} = 1 \]
This simplifies to:
\[ x - \frac{ky}{9} = 1 \]

### Step 4: Rearranging the equation
Rearranging the equation gives:
\[ x - 1 = \frac{ky}{9} \]
Multiplying through by 9 leads to:
\[ 9(x - 1) = ky \]
Thus, we can express this as:
\[ ky - 9x + 9 = 0 \]

### Step 5: Find the corresponding pair of tangents
To find the corresponding pair of tangents, we need to eliminate \( k \) from the equation. The general form of the equation of the tangents to the hyperbola is:
\[ S = T^2 \]
where \( S = x^2 - y^2 - 9 \) and \( T = kx - 9y + 9 \).

Setting \( S = T^2 \):
\[ (x^2 - y^2 - 9)(k^2 - 18y + 81) = 0 \]
This leads to a quadratic equation in \( k \), which can be solved to find the values of \( k \).

### Step 6: Final equation
After solving the quadratic equation, we can find the coefficients and write the final equation of the pair of tangents.

The final equation of the pair of tangents to the hyperbola \( H \) from the line \( L \) is:
\[ 9x^2 - 8y^2 - 18x + 9 = 0 \]