Question

If the equation $3x^2 - kxy - 3y^2 = 0$ represents the bisectors of angles between the lines $x^2 - 3xy - 4y^2 = 0$, then value of $k$ is

• A. $-6$
• B. $-10$
• C. $6$
• D. $10$

Hint:

The equation of the bisectors $\frac{x^2 - y^2}{a - b} = \frac{xy}{h}$ is a very important standard result for pair of straight lines. Always directly compare the coefficients of your derived equation with the given one to avoid sign errors.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the joint equation of a pair of straight lines and the joint equation of their angle bisectors (with an unknown parameter $k$). We need to find the value of $k$.

Step 2: Key Formula or Approach:
For a pair of lines given by $ax^2 + 2hxy + by^2 = 0$, the joint equation of their angle bisectors is given by the standard formula:
$$\frac{x^2 - y^2}{a - b} = \frac{xy}{h}$$

Step 3: Detailed Explanation:
The given equation for the original pair of lines is $x^2 - 3xy - 4y^2 = 0$.
Comparing this with $ax^2 + 2hxy + by^2 = 0$, we get:
$a = 1$
$2h = -3 \implies h = -\frac{3}{2}$
$b = -4$
Substitute these values into the angle bisector formula:
$$\frac{x^2 - y^2}{1 - (-4)} = \frac{xy}{-\frac{3}{2}}$$ $$\frac{x^2 - y^2}{5} = \frac{2xy}{-3}$$ Cross-multiply to simplify:
$$-3(x^2 - y^2) = 10xy$$ $$-3x^2 + 3y^2 = 10xy$$ Rearrange the terms to match the format of the given bisector equation:
$$3x^2 + 10xy - 3y^2 = 0$$ The problem states that the bisector equation is $3x^2 - kxy - 3y^2 = 0$.
Comparing the coefficient of the $xy$ term in both equations:
$$-k = 10 \implies k = -10$$

Step 4: Final Answer:
The value of $k$ is $-10$, matching option (B).