Question

If one of the lines given by $kx^2 + xy - y^2 = 0$ bisects the angle between the co-ordinate axes, then the values of $k$ are

• A. $1$ and $2$
• B. $0$ and $2$
• C. $0$ and $-2$
• D. $-1$ and $2$

Hint:

Whenever a question states that a line "bisects the coordinate axes," immediately substitute $y = x$ and $y = -x$ directly into the given expression. For $y=x$: $kx^2 + x^2 - x^2 = 0 \implies kx^2 = 0 \implies k = 0$. For $y=-x$: $kx^2 - x^2 - x^2 = 0 \implies (k-2)x^2 = 0 \implies k = 2$. This direct substitution approach takes only seconds!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The equation $kx^2 + xy - y^2 = 0$ represents a pair of straight lines passing through the origin. We are told that one of these constituent lines bisects the angle between the coordinate axes, and we need to determine the possible values for the parameter $k$.

Step 2: Key Formula or Approach:
A straight line that bisects the angle between the coordinate axes makes an angle of either $45^\circ$ or $135^\circ$ with the positive $X$-axis. This leaves two possible values for its slope $m$: $$m = \tan(45^\circ) = 1 \quad \text{or} \quad m = \tan(135^\circ) = -1 \implies m = \pm 1$$ Since the line passes through the origin, its equation is $y = mx \implies \frac{y}{x} = m$. We can substitute $\frac{y}{x} = \pm 1$ directly into our homogeneous line pair equation to isolate $k$.

Step 3: Detailed Explanation:
Divide the given equation $kx^2 + xy - y^2 = 0$ by $x^2$ to express it in terms of the slope ratio $\frac{y}{x}$: $$k + \left(\frac{y}{x}\right) - \left(\frac{y}{x}\right)^2 = 0$$ Substitute the slope property $\frac{y}{x} = m$ into this auxiliary layout: $$k + m - m^2 = 0$$ Now substitute our two possible values for $m$ into this equation to solve for $k$:

• Case 1: If $m = 1$ $$k + 1 - (1)^2 = 0 \implies k + 1 - 1 = 0 \implies k = 0$$

• Case 2: If $m = -1$ $$k + (-1) - (-1)^2 = 0 \implies k - 1 - 1 = 0 \implies k - 2 = 0 \implies k = 2$$
This yields two possible values for $k$: $0$ and $2$, which perfectly matches option (B).

Step 4: Final Answer:
The possible values of $k$ are $0$ and $2$, which corresponds to option (B).