JEE Main 2025 Apr 3 Shift 1, Mathematics Q13: Points and Straight Lines

JEE Main2025Apr 3, Shift 1Mathematics

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $L_{1} : 2 x + y + 6 = 0$ and $L_{2} : 4 x + 2 y - p = 0$ , $p > 0$ , at the points $A$ and $B$ , respectively. If $A B = \frac{9}{2}$ and the foot of the perpendicular from the point $A$ on the line $L_{2}$ is $M$ , then $\frac{A M}{B M}$ is equal to

Solution

A line through the origin making equal angles with the positive axes (in 2D) has direction $(1, 1)$ , i.e. the line $y = x .$

In the right triangle $A M B$ (right angle at $M$ , since $M$ is the foot of perpendicular from $A$ to $L_{2}$ ), the angle at $B$ equals the angle between the transversal line $y = x$ and the line $L_{2} : 4 x + 2 y - p = 0$ (slope $- 2$ ).

Slopes: $m_{1} = 1$ , $m_{2} = - 2$ , so

$tan θ = \frac{1 - ( - 2 )}{1 + ( 1 ) ( - 2 )} = \frac{3}{- 1} = 3.$

From the right triangle, $tan θ = \frac{A M}{B M} = 3.$

Concept behind this question

Points and Straight LinesNotes, formulas and examples →

Practice more Mathematics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →

JEE Main 2025 Apr 3 Shift 1, Mathematics Q13: Points and Straight Lines

Concept behind this question

More previous year questions on Points and Straight Lines

Practice more Mathematics