Question

If the plane $2x-3y+6z-11=0$ makes an angle $\theta$ with the $x-axis$, then the value of $\tan \theta$ is equal to

• A. $\frac{2}{3}$
• B. $\frac{2}{15}$
• C. $\frac{\sqrt{2}}{3}$
• D. $\frac{2\sqrt{5}}{15}$

Answer 1

Answer: (D)

$\frac{2\sqrt{5}}{15}$

Answer 2

Let the equation of X-axis be $\frac{x-0}{a}=\frac{y-0}{0}=\frac{z-0}{0}$ Here, ${{a}_{2}}=a,\,\,\,{{b}_{1}}=0$ and ${{c}_{1}}=0$ Given equation of plane is $2x-3y+6z-11=0$ Here, ${{a}_{2}}=2,\,\,{{b}_{2}}=-3,\,{{c}_{2}}=6$
Let $\theta$ be the angle between line and plane. Then,
$\sin \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\,\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}$
$=\frac{a\times 2+0\times (-3)+0\times 6}{\sqrt{{{a}^{2}}+{{0}^{2}}+{{0}^{2}}}\,\sqrt{{{2}^{2}}+{{(-3)}^{2}}+{{6}^{2}}}}$
$=\frac{2a}{\sqrt{{{a}^{2}}}\sqrt{4+9+36}}=\frac{2a}{a.\,7}=\frac{2}{7}$
$\Rightarrow$ $\sin \,\,\theta =\frac{2}{7}$