Question

Assertion (A) :The angle between the pair of planes $2x^2-6y^2-12z^2+18yz+2zx+xy=0$ is $co{s}^{-1}\left(\pm \frac{16}{21}\right)$
Reason (R) :The second degree equation in three variables $x,y,z$ given by $a{x}^2+b{y}^2+c{z}^2+2fyz+2gzx+2hxy=0$ represent a pair of planes if $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A)
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A)
• C. (A) is true but (R) is false
• D. (A) is false but (R) is true

Answer 1

Answer: (B)

The terms of $x$ & $y$ i.e. $2x^2+xy-6y^2$ can be factorize as $(x+2y)(2x-3y)$ Now , assume the factors to the given pair of planes are $(x+2y+pz),(2x-3y+qz)$
$\therefore 2x^2-6y^2-12z^2+18yz+2zx+xy$
$=(x+2y+pz)(2x-3y+qz)$
$\Rightarrow 2p+q=2\dots (i)$
and $-3p+2q=18\dots (ii)$
Also, $pq=-12\dots (iii)$
Solving (i) & (ii), we get $P=-2,q=6$ which satisfies $\dots (iii)$
$\therefore$ Equation of two planes are $x+2y-2z=0$ & $2x-3y+6z=0$
If $\theta$ be the angle between the planes
$\therefore cos\theta =\pm \frac{1\left(2\right)+2\left(-3\right)+\left(-2\right)\left(6\right)}{\sqrt{1+4+4}\sqrt{4+9+36}}$
$\Rightarrow cos\theta =\pm \frac{16}{21}$
$\therefore \theta =co{s}^{-1}\left(\frac{16}{21}\right)$
$\therefore$ Assertion (A) is true & Reason (R) is obvious but not the proper explanation for Assertion (A)