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A line makes intercepts 5 and 7 on the coordinate axes. The axes are rotated through an angle $\theta$ in the positive direction about the origin so that the line makes equal intercepts on the new axes, $\operatorname{then}|\tan \theta|=$
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Verified Answer
The correct answer is:
$\frac{1}{6}$
Given intercepts oncoordinate axes are 5 and 7
Let $\mathrm{a}=5, \mathrm{~b}=7$, then equation of line is
$$
\begin{aligned}
& \frac{x}{5}+\frac{y}{7}=1 \\
& 7 x+5 y=35
\end{aligned}
$$
According to question, the equation of new line is $\mathrm{x}$
$$
\mathrm{x}(\mathrm{b} \cos \theta+\mathrm{a} \sin \theta)+\mathrm{y}(\mathrm{a} \cos \theta-\mathrm{b} \sin \theta)=\mathrm{ab}
$$
$$
\frac{\frac{\mathrm{x}}{\mathrm{ab}}}{\mathrm{b} \cos \theta+\mathrm{a} \sin \theta}+\frac{\frac{\mathrm{y}}{-\mathrm{ab}}}{-\mathrm{b} \sin \theta+\mathrm{a} \cos \theta}=1
$$
so, $\mathrm{b} \cos \theta+\mathrm{a} \sin \theta=\mathrm{a} \cos \theta-\mathrm{b} \sin \theta$
$$
7 \cos \theta+5 \sin \theta=5 \cos \theta-7 \sin \theta
$$
$$
12 \sin \theta=2 \cos \theta
$$
$$
\tan \theta=\frac{-2}{12}=\frac{-1}{6} \Rightarrow|\tan \theta|=\frac{1}{6}
$$
So, option (b) is correct.
Let $\mathrm{a}=5, \mathrm{~b}=7$, then equation of line is
$$
\begin{aligned}
& \frac{x}{5}+\frac{y}{7}=1 \\
& 7 x+5 y=35
\end{aligned}
$$
According to question, the equation of new line is $\mathrm{x}$
$$
\mathrm{x}(\mathrm{b} \cos \theta+\mathrm{a} \sin \theta)+\mathrm{y}(\mathrm{a} \cos \theta-\mathrm{b} \sin \theta)=\mathrm{ab}
$$
$$
\frac{\frac{\mathrm{x}}{\mathrm{ab}}}{\mathrm{b} \cos \theta+\mathrm{a} \sin \theta}+\frac{\frac{\mathrm{y}}{-\mathrm{ab}}}{-\mathrm{b} \sin \theta+\mathrm{a} \cos \theta}=1
$$
so, $\mathrm{b} \cos \theta+\mathrm{a} \sin \theta=\mathrm{a} \cos \theta-\mathrm{b} \sin \theta$
$$
7 \cos \theta+5 \sin \theta=5 \cos \theta-7 \sin \theta
$$
$$
12 \sin \theta=2 \cos \theta
$$
$$
\tan \theta=\frac{-2}{12}=\frac{-1}{6} \Rightarrow|\tan \theta|=\frac{1}{6}
$$
So, option (b) is correct.
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