The point of contact of 9x+8y-11=0 to the hyperbola `3x^(2)-4y^(2)=11` is

Options

A. (3,-2)
B. (3,2)
C. (-3,-3)
D. (3,3)

Answer

To find the point of contact of the line \(9x + 8y - 11 = 0\) with the hyperbola \(3x^2 - 4y^2 = 11\), we can follow these steps:

### Step 1: Write down the equations
The equations we have are:
1. Hyperbola: \(3x^2 - 4y^2 = 11\) (Equation 1)
2. Line: \(9x + 8y - 11 = 0\) (Equation 2)

### Step 2: Solve the line equation for \(y\)
From Equation 2, we can express \(y\) in terms of \(x\):
\[ 8y = 11 - 9x \implies y = \frac{11 - 9x}{8} \]
Let this be Equation 3.

### Step 3: Substitute \(y\) in the hyperbola equation
Now, substitute Equation 3 into Equation 1:
\[ 3x^2 - 4\left(\frac{11 - 9x}{8}\right)^2 = 11 \]

### Step 4: Simplify the equation
First, calculate \(\left(\frac{11 - 9x}{8}\right)^2\):
\[ \left(\frac{11 - 9x}{8}\right)^2 = \frac{(11 - 9x)^2}{64} = \frac{121 - 198x + 81x^2}{64} \]
Now substitute this back into the hyperbola equation:
\[ 3x^2 - 4\left(\frac{121 - 198x + 81x^2}{64}\right) = 11 \]
Multiply through by 64 to eliminate the fraction:
\[ 192x^2 - 4(121 - 198x + 81x^2) = 704 \]
Expanding gives:
\[ 192x^2 - 484 + 792x - 324x^2 = 704 \]
Combine like terms:
\[ (192 - 324)x^2 + 792x - 484 - 704 = 0 \]
This simplifies to:
\[ -132x^2 + 792x - 1188 = 0 \]
Dividing the entire equation by -12:
\[ 11x^2 - 66x + 99 = 0 \]

### Step 5: Solve the quadratic equation
Now we can solve this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 11\), \(b = -66\), and \(c = 99\):
\[ x = \frac{66 \pm \sqrt{(-66)^2 - 4 \cdot 11 \cdot 99}}{2 \cdot 11} \]
Calculating the discriminant:
\[ (-66)^2 - 4 \cdot 11 \cdot 99 = 4356 - 4356 = 0 \]
Since the discriminant is 0, there is one real solution:
\[ x = \frac{66}{22} = 3 \]

### Step 6: Find \(y\) using the value of \(x\)
Substituting \(x = 3\) back into Equation 3 to find \(y\):
\[ y = \frac{11 - 9 \cdot 3}{8} = \frac{11 - 27}{8} = \frac{-16}{8} = -2 \]

### Step 7: Conclusion
The point of contact is:
\[ (3, -2) \]