Question

If the line $\overline{OR}$ makes angles $\theta_1,\theta_2,\theta_3$ with the planes $XOY, YOZ, ZOX$ respectively, $cos^2\, \theta_1+cos^2\,\theta_2+\,cos^3 \theta_3$ =

• 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 $ $\sin^2 \, \theta_1 +\sin^2 \, \theta_2 + \sin^2 \, \theta_3$
= $\frac{x^2 + y^2 +z^2}{x^2 + y^2 + z^2} = 1$
$\Rightarrow 1 - \cos^2 \, \theta_1 + 1 - \cos^2 \, \theta_2 + 1 - \cos^2 \, \theta_3 = 1$
$\cos^2 \, \theta_1 + \cos^2 \, \theta_2 + \cos^2 \, \theta_3 = 3 - 1 = 2$