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 :

JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Straight Lines

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 :

(A) $\frac{8}{5}$
(B) $\frac{25}{41}$
(C) $\frac{2}{5}$
(D) $\frac{30}{41}$ ✓ Correct

Answer: (D)

Solution

The line

4 x + 5 y = 20

meets the axes at

(5, 0)

and

(0, 4)

. Trisecting this segment gives

A = (\frac{10}{3}, \frac{4}{3})

and

B = (\frac{5}{3}, \frac{8}{3})

.

Slopes from the origin:

m_{1} = \frac{4/3}{10/3} = \frac{2}{5}

m_{2} = \frac{8/3}{5/3} = \frac{8}{5}

tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}} = \frac{8/5 - 2/5}{1 + 16/25} = \frac{6/5}{41/25} = \frac{30}{41}

JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Single correct option

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