Question

The acute angle between the planes $P _{1}$ and $P _{2}$, when $P_{1}$ and $P_{2}$ are the planes passing through the intersection of the planes $5 x+8 y+13 z-29=0$ and $8 x -7 y + z -20=0$ and the points $(2,1,3)$ and $(0,1,2)$, respectively, is

• A. $\frac{\pi}{3}$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{6}$
• D. $\frac{\pi}{12}$

Answer 1

Answer: (A)

Equation of plane passing through the intersection of planes
$5 x+8 y+13 z-29=0$ and $8 x-7 y+z-$ $20=0$ is
$5 x+8 y+3 z-29+\lambda(8 x-7 y+z-20)=0$ and if it is passing through $(2,1,3)$ then $\lambda=\frac{7}{2}$
$P_{1}$ : Equation of plane through intersection of $5 x+8 y+13 z-29=0$ and $8 x-7 y+z-20=0$ and the point $(2,1,3)$ is
$5 x+8 y+3 z-29+\frac{7}{2}(8 x-7 y+z-20)=0$
$\Rightarrow 2 x-y+z=6$
Similarly $P _{2}$ : Equation of plane through intersection of
$5 x+8 y+13 z-29=0$ and $8 x-7 y+z-20=0$ and the point $(0,1,2)$ is
$\Rightarrow x+y+2 z=5$
Angle between planes $=\theta=\cos ^{-1}\left(\frac{3}{\sqrt{6} \sqrt{6}}\right)=\frac{\pi}{3}$