Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
A line '$L$' passes through the point $(2,3)$ and making intercept of length $3$ units between the lines $2x + y - 2 = 0$ and $4x + 2y - 10 = 0$ then which of the following may be true about the line $L$:
Parallel to $x$ axis
Parallel to $y$-axis
having slope $-\frac{3}{4}$
Perpendicular to $2x + y - 2 = 0$
Step-by-Step Solution
Key Concept: The angle between two lines with slopes $m_1$ and $m_2$ is given by $\tan \theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|$.
Given that the slope of line $L$ is $m$, we use the angle formula $\tan \alpha = \frac{m-(-2)}{1+m(-2)} = \frac{1}{2}$. This simplifies to $2(m+2) = \pm(1-2m)$, giving $4m = -3$ or $m = -1$ (perpendicular to x-axis). The two solutions are $m = -\frac{3}{4}$ or the line is vertical.
Correct Answer: 2,3