If `x=9` is the chord of contact of the hyperbola `x^2-y^2=9` then the equation of the corresponding pair of tangents is (A) `9x^2-8y^2+18x-9=0` (B) `9x^2-8y^2-18x+9=0` (C) `9x^2-8y^2-18x-9=0` (D) `9x^2-8y^2+18x+9=0``

Options

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

Explanation

To find the equation of the corresponding pair of tangents to the hyperbola \(x^2 - y^2 = 9\) given that \(x = 9\) is the chord of contact, we can follow these steps:

### Step 1: Identify the hyperbola and the chord of contact
The equation of the hyperbola is given as:
\[ x^2 - y^2 = 9 \]
The chord of contact is given by the line \(x = 9\).

### Step 2: Find the points of intersection
To find the points where the line \(x = 9\) intersects the hyperbola, substitute \(x = 9\) into the hyperbola's equation:
\[ 9^2 - y^2 = 9 \]
This simplifies to:
\[ 81 - y^2 = 9 \]
Rearranging gives:
\[ y^2 = 81 - 9 = 72 \]
Taking the square root, we find:
\[ y = \pm 6\sqrt{2} \]
Thus, the points of intersection are:
\[ (9, 6\sqrt{2}) \quad \text{and} \quad (9, -6\sqrt{2}) \]

### Step 3: Use the points to find the equations of the tangents
The general formula for the equation of the tangent to the hyperbola \(x^2 - y^2 = a^2\) at the point \((x_1, y_1)\) is:
\[ xx_1 - yy_1 = a^2 \]
In our case, \(a^2 = 9\), and we will consider both points of contact.

#### For the point \(A(9, 6\sqrt{2})\):
Substituting \(x_1 = 9\) and \(y_1 = 6\sqrt{2}\):
\[ 9x - 6\sqrt{2}y = 9 \]
Rearranging gives:
\[ 9x - 6\sqrt{2}y - 9 = 0 \]
Dividing through by 3:
\[ 3x - 2\sqrt{2}y - 3 = 0 \quad \text{(Equation 1)} \]

#### For the point \(B(9, -6\sqrt{2})\):
Substituting \(x_1 = 9\) and \(y_1 = -6\sqrt{2}\):
\[ 9x + 6\sqrt{2}y = 9 \]
Rearranging gives:
\[ 9x + 6\sqrt{2}y - 9 = 0 \]
Dividing through by 3:
\[ 3x + 2\sqrt{2}y - 3 = 0 \quad \text{(Equation 2)} \]

### Step 4: Multiply the equations to find the pair of tangents
Now we will multiply the two equations (Equation 1 and Equation 2):
\[ (3x - 2\sqrt{2}y - 3)(3x + 2\sqrt{2}y - 3) = 0 \]
Using the difference of squares:
\[ (3x - 3)^2 - (2\sqrt{2}y)^2 = 0 \]
Expanding this gives:
\[ 9(x - 1)^2 - 8y^2 = 0 \]
This can be rearranged to:
\[ 9x^2 - 18x + 9 - 8y^2 = 0 \]
Thus, the equation of the corresponding pair of tangents is:
\[ 9x^2 - 8y^2 - 18x + 9 = 0 \]

### Final Answer
The correct option is:
**(B) \(9x^2 - 8y^2 - 18x + 9 = 0\)**

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