CUET PG 2022 — Mathematics PYQ

Step 1: Understand the Bisector Line

The line that bisects the angle between the axes in the first quadrant is the line making an angle of $45^{\circ }$ with the positive $x$-axis.

The equation of this line is:

Or, rewritten as:

Step 2: Use the Homogeneous Equation Property

The given pair of straight lines is:

If we divide the entire equation by $x^2$ (where $x\ne 0$), we get:

Let $m=\frac{y}{x}$ be the slope of the lines. The auxiliary equation is:

Step 3: Substitute the Condition

Since one of the lines is the bisector $y=x$, its slope is $m=1$. This value of $m$ must satisfy the auxiliary equation:

Rearranging the terms, we get the required condition:

Step 4: Alternative Case (Second/Fourth Quadrant Bisector)

While the question specifies the first quadrant, if the line bisected the axes in the second quadrant ($y=-x$), the slope would be $m=-1$. Substituting that would give:

Combining both possibilities (if the question meant any angle bisector), the general condition is often written as: