Question

Equation of a straight line passing through the origin and making with $x$ - axis an angle twice the size of the angle made by the line $y=0.2x$ with the $x$ - axis, is :

• A. $y=4.0x$
• B. $y=\frac{5}{12}x$
• C. $6y-5y=0$
• D. $noneofthese$

Answer 1

The correct option is B $y=\frac{5}{12}x$

Since the line passes through the origin, its equation will be like: $y=mx$

(m = slope of the line)

Angle made by line y = $0.2x$ will be ${\tan }^{-1}\mathrm{0.2.}$

Angle made by required line = twice of above angle $=2{\tan }^{-1}0.2$

Slope of our line $=\tan \theta =\tan \left(2ta{n}^{-1}0.2\right)$.

Using the property $\tan 2x=\frac{2\tan x}{1-{\tan }^{2}x}$

slope $=\frac{2\tan \left({\tan }^{-1}0.2\right)}{(1-ta{n}^2({\tan }^{-1}0.2\left)\right)}$

Now using the property: $\tan \left({\tan }^{-1}x\right)=x$

slope $=\frac{2\left(0.2\right)}{(1-{0.2}^2)}$

$=\frac{0.4}{0.96}$

$=0.416$

Therefore, Equation of our required line:

$y=\frac{5}{12}x$