JEE MAIN - Mathematics (2024 - 27th January Morning Shift - No. 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 :
$\frac{30}{41}$
$\frac{8}{5}$
$\frac{2}{5}$
$\frac{25}{41}$
Explanation
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 $$\mathrm{OB}=\mathrm{m}_2=\frac{2}{5}$$
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$$\begin{aligned} & \tan \theta=\left|\frac{m_1-m_2}{1+m_1 m_2}\right| \\ & \tan \theta=\frac{\frac{6}{5}}{1+\frac{16}{25}}=\frac{30}{41} \\ & \tan \theta=\frac{30}{41} \end{aligned}$$
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