Question

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

• 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
$\therefore \, cos\,\theta = \pm \frac{1\left(2\right)+2\left(-3\right)+\left(-2\right)\left(6\right)}{\sqrt{1+4+4}\sqrt{4+9+36}}$
$\Rightarrow cos\,\theta=\pm\,\frac{16}{21}$
$\therefore \theta=cos^{-1}\left(\frac{16}{21}\right)$
$\therefore $ Assertion (A) is true & Reason (R) is obvious but not the proper explanation for Assertion (A)