Question

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

• A. 1, 2
• B. 1, 3
• C. 0, 2
• D. -2, 2

Hint:

For angle bisectors, use the condition that the slope of the line is equal to the tangent of half the angle between the axes.

Answer 1

The Correct Option is C

Step 1: Equation of the line.
We are given the equation \( kx^2 + xy - y^2 = 0 \). This can be rearranged as: \[ y = \frac{kx^2}{x - y} \] To find the value of \( k \), we use the condition that the line bisects the angle between the co-ordinate axes. Step 2: Condition for angle bisector.
The condition for a line to bisect the angle between the axes is that the slope of the line is equal to the tangent of half the angle between the axes. For the co-ordinate axes, this angle is \( 45^\circ \), so the slope is 1. Using this condition and solving the equation, we get the values of \( k \) as \( 0 \) and \( 2 \). Step 3: Conclusion.
Thus, the values of \( k \) are \( \boxed{0, 2} \).