Question

Assertion :The angle between the pair of planes $2x^2-6y^2-12z^2+18yz+2zx+xy=0$ is ${\cos }^{-1}(\pm \frac{16}{21})$ Reason: 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 Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is B Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion

Reason:

$a{x}^2+b{y}^2+c{z}^2+2fyz+2gzx+2hxy=0$ represent a plane of $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$ is correct since it is given by the matrix form. But it does not gives the formula between angle and the plane.

Assertion: $2x^2-6y^2-12z^2+18yz+2zx+xy=0$

$(x+2y-2z)(2x-3y+6z)=0$

so $⟨1,2,-2⟩,⟨2,-3,6⟩$

Angle $\theta =\cos \theta =\frac{2-6-12}{\sqrt{9}\times \sqrt{49}}=\frac{-16}{3\times 7}=\frac{-16}{21}$

Thus A is also correct but reason does not explain A