Which of the following is correct to the line 5x+y+3z=0 and 3x−y+z+1=0 - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

A
Symmetrical form of the equation of line is $\frac{x}{2} = \frac{y - \frac{1}{8}}{- 1} = \frac{z + \frac{5}{8}}{1}$
B
Symmetrical form of the equations of line is $\frac{x + \frac{1}{8}}{1} = \frac{y - \frac{5}{8}}{1} = \frac{z}{- 2}$
C
Equation of the plane through $(2, - 1, 4)$ and perpendicular to the given line is $2 x - y + z - 7 = 0$
D
Equation of line through $(2, - 1, 4)$ and perpendicular to the given line is $x + y - 2 z + 5 = 0$

The given planes are $5 x + y + 3 z = 0$ and $3 x - y + z + 1 = 0$

Suppose that the direction ratios of line of intersection of the above two planes is $⟨ a, b, c ⟩$

Hence, $3 a - b + c = 0, 5 a + b + 3 c = 0$ $\Rightarrow \frac{a}{1} = \frac{b}{1} = \frac{c}{- 2}$

To get a point on the line, substitute $z = 0$ in the plane equations and the solve for the other variables

hence a point on the line of intersection of planes is $P (- \frac{1}{8}, \frac{5}{8}, 0)$

Therefore, the equation of the required line is $\frac{x + \frac{1}{8}}{1} = \frac{y - \frac{5}{8}}{1} = \frac{z}{- 2}$

Which of the following is correct to the line $5 x + y + 3 z = 0$ and $3 x - y + z + 1 = 0$