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 $5x+8y+13z-29=0$ and $8x-7y+z-20=0$ 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
$5x+8y+13z-29=0$ and $8x-7y+z-$ $20=0$ is
$5x+8y+3z-29+\lambda (8x-7y+z-20)=0$ and if it is passing through $(2,1,3)$ then $\lambda =\frac{7}{2}$
$P_1$ : Equation of plane through intersection of $5x+8y+13z-29=0$ and $8x-7y+z-20=0$ and the point $(2,1,3)$ is
$5x+8y+3z-29+\frac{7}{2}(8x-7y+z-20)=0$
$\Rightarrow 2x-y+z=6$
Similarly $P_2$ : Equation of plane through intersection of
$5x+8y+13z-29=0$ and $8x-7y+z-20=0$ and the point $(0,1,2)$ is
$\Rightarrow x+y+2z=5$
Angle between planes $=\theta ={\cos }^{-1}\left(\frac{3}{\sqrt{6}\sqrt{6}}\right)=\frac{\pi }{3}$