The portion of the line 4 x+5 y=20 in the first quadrant is trisected by the lines L 1 and L 2 passing through the origin. The tangent of an angle between the lines L 1 and L 2 is :

Mathematics · Straight lines

JEE Main 2024 — 27 January, Shift 1 — Question 14

The portion of the line $4 x+5 y=20$ in the first quadrant is trisected by the lines $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ passing through the origin. The tangent of an angle between the lines $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ is :

Option A:
$\frac{8}{5}$
Option B:
$\frac{25}{41}$
Option C:
$\frac{2}{5}$
Option D:
$\frac{30}{41}$
Correct

Answer: D

Step-by-step solution

Co-ordinates of $\mathrm{A}=\left(\frac{5}{3}, \frac{8}{3}\right)$

Co-ordinates of $\mathrm{B}=\left(\frac{10}{3}, \frac{4}{3}\right)$

Slope of $\mathrm{OA}=\mathrm{m}_{1}=\frac{8}{5}$

Slope of $\mathrm{OB}=\mathrm{m}_{2}=\frac{2}{5}$

$\tan \theta=\left|\frac{\mathrm{m}_{1}-\mathrm{m}_{2}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\right|$

$\tan \theta=\frac{\frac{6}{5}}{1+\frac{16}{25}}=\frac{30}{41}$

$\tan \theta=\frac{30}{41}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 27 January, Shift 1
Subject: Mathematics
Chapter: Straight lines
Topic: Angle between lines, perpendicular distance & distance between parallel lines, foot, image