Question

If the line $\overline{OR}$ makes angles ${\theta }_1,{\theta }_2,{\theta }_3$ with the planes $XOY,YOZ,ZOX$ respectively, $co{s}^2{\theta }_1+co{s}^2{\theta }_2+co{s}^3{\theta }_3$ =

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

Let $P(x,y,z)$ be any point on the line OP, D-Ratio of OP are $x,y,z.$ D.R. of normal of $XOY$ plane $i.e.,z$ = 0 are 0, 0, 1
$\therefore$ $\sin {\theta }_1=\frac{x.0+y.0+z.1}{\sqrt{x^2+y^2+z^2}}$
= $\frac{z}{\sqrt{x^2+y^2+z^2}}$
Similarly $\sin {\theta }_2=\frac{y}{\sqrt{x^2+y^2+z^2}}$
$\sin {\theta }_3=\frac{x}{\sqrt{x^2+y^2+z^2}}$
$\therefore$ ${\sin }^2{\theta }_1+{\sin }^2{\theta }_2+{\sin }^2{\theta }_3$
= $\frac{x^2+y^2+z^2}{x^2+y^2+z^2}=1$
$\Rightarrow 1-{\cos }^2{\theta }_1+1-{\cos }^2{\theta }_2+1-{\cos }^2{\theta }_3=1$
${\cos }^2{\theta }_1+{\cos }^2{\theta }_2+{\cos }^2{\theta }_3=3-1=2$