From a point on the line `x-y+2-0` tangents are drawn to the hyperbola `(x^(2))/(6)-(y^(2))/(2)=1` such that the chord of contact passes through a fixed point `(lambda, mu)`. Then, `mu-lambda` is equal to

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to find the value of \( \mu - \lambda \) given the conditions of the tangents drawn from a point on the line \( x - y + 2 = 0 \) to the hyperbola \( \frac{x^2}{6} - \frac{y^2}{2} = 1 \).

### Step 1: Identify the point on the line
The line equation is given as \( x - y + 2 = 0 \). We can express \( y \) in terms of \( x \):
\[ y = x + 2 \]
Let the coordinates of the point on the line be \( (x_1, y_1) \). Substituting \( y_1 \):
\[ y_1 = x_1 + 2 \]

### Step 2: Write the equation of the chord of contact
For the hyperbola \( \frac{x^2}{6} - \frac{y^2}{2} = 1 \), the equation of the chord of contact from the point \( (x_1, y_1) \) is given by:
\[ \frac{xx_1}{6} - \frac{yy_1}{2} = 1 \]
Substituting \( y_1 = x_1 + 2 \):
\[ \frac{xx_1}{6} - \frac{y(x_1 + 2)}{2} = 1 \]

### Step 3: Simplify the chord of contact equation
Multiplying through by 6 to eliminate the denominators:
\[ xx_1 - 3y(x_1 + 2) = 6 \]
Expanding this gives:
\[ xx_1 - 3yx_1 - 6y = 6 \]
Rearranging terms:
\[ xx_1 - 3yx_1 - 6y - 6 = 0 \]
Factoring out \( x_1 \):
\[ x_1(x - 3y) - 6y - 6 = 0 \]

### Step 4: Determine the fixed point condition
For the chord of contact to pass through a fixed point \( (\lambda, \mu) \), we substitute \( x = \lambda \) and \( y = \mu \):
\[ x_1(\lambda - 3\mu) - 6\mu - 6 = 0 \]
This implies:
\[ x_1(\lambda - 3\mu) = 6\mu + 6 \]

### Step 5: Analyze the relationship
For the equation to hold for all \( x_1 \), the coefficient of \( x_1 \) must be zero:
\[ \lambda - 3\mu = 0 \]
Thus, we have:
\[ \lambda = 3\mu \]

### Step 6: Substitute to find \( \mu - \lambda \)
Now, substituting \( \lambda = 3\mu \) into \( \mu - \lambda \):
\[ \mu - \lambda = \mu - 3\mu = -2\mu \]

### Step 7: Find specific values
From the previous analysis, we can set \( \mu = -1 \) (as suggested in the transcript), then:
\[ \lambda = 3(-1) = -3 \]
Calculating \( \mu - \lambda \):
\[ \mu - \lambda = -1 - (-3) = -1 + 3 = 2 \]

### Final Answer
Thus, the value of \( \mu - \lambda \) is:
\[ \boxed{2} \]