If x= 9 is a chord of contact of the hyperbola ` x^(2) -y^(2) =9`, then the equation of the tangents at one of the points of contact is

Options

A. a) `x+sqrt3 y+2=0`
B. b) `3x-2sqrt2 y-3=0`
C. c) `3x-sqrt2 y+6=0`
D. d) `x-sqrt3 y+2=0`

Answer

To solve the problem step by step, we need to find the equation of the tangents at one of the points of contact for the hyperbola given by the equation \(x^2 - y^2 = 9\), where \(x = 9\) is a chord of contact.

### Step 1: Identify the hyperbola equation
The hyperbola is given by:
\[ x^2 - y^2 = 9 \]

### Step 2: Substitute \(x = 9\) into the hyperbola equation
Since \(x = 9\) is a chord of contact, we substitute \(x = 9\) into the hyperbola equation:
\[ 9^2 - y^2 = 9 \]
This simplifies to:
\[ 81 - y^2 = 9 \]

### Step 3: Solve for \(y^2\)
Rearranging the equation gives:
\[ y^2 = 81 - 9 = 72 \]

### Step 4: Find the values of \(y\)
Taking the square root of both sides, we find:
\[ y = \pm \sqrt{72} = \pm \sqrt{36 \cdot 2} = \pm 6\sqrt{2} \]
Thus, the points of contact are \( (9, 6\sqrt{2}) \) and \( (9, -6\sqrt{2}) \).

### Step 5: Write the equation of the tangent
The general equation of the tangent to the hyperbola \(x^2 - y^2 = a^2\) at the point \((x_1, y_1)\) is given by:
\[ x \cdot x_1 - y \cdot y_1 = a^2 \]
Here, \(a^2 = 9\), and we can take one of the points of contact, say \((9, 6\sqrt{2})\):
\[ x \cdot 9 - y \cdot 6\sqrt{2} = 9 \]

### Step 6: Simplify the tangent equation
Rearranging gives:
\[ 9x - 6\sqrt{2}y = 9 \]
Dividing the entire equation by 3 to simplify:
\[ 3x - 2\sqrt{2}y = 3 \]

### Final Answer
Thus, the equation of the tangent at one of the points of contact is:
\[ 3x - 2\sqrt{2}y = 3 \]

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