Question

The equation of plane through the line of intersection of planes $ax + by + cz + d = 0$, $a'x + b'y + c'z + d' = 0$ and parallel to the line $y = 0,z = 0$ is

• A. $(ab' - a'b)x + (bc' - b'c)y + (ad' - a'd) = 0$
• B. $(ab' - a'b)x + (bc' - b'c)y + (ad' - a'd)z = 0$
• C. $(ab' - a'b)y + (ac' - a'c)z + (ad' - a'd) = 0$
• D. None of these

Answer 1

Answer: (C)

$(ab' - a'b)y + (ac' - a'c)z + (ad' - a'd) = 0$

Answer 2

The equation of a plane through the line of intersection of the planes $a x + b y + c z + d = 0$ and $a ^ { \prime } x + b ^ { \prime } y + c ^ { \prime } z + d ^ { \prime } = 0$ is

$( a x + b y + c z + d ) + \lambda \left( a ^ { \prime } x + b ^ { \prime } y + c ^ { \prime } z + d ^ { \prime } \right) = 0$

or$x \left( a + \lambda a ^ { \prime } \right) + y \left( b + \lambda b ^ { \prime } \right) + z \left( c + \lambda c ^ { \prime } \right) + d + \lambda d ^ { \prime } = 0$ ….(i)

This is parallel to x-axis i.e., $y = 0 , z = 0$

$\therefore 1 \left( a + \lambda a ^ { \prime } \right) + 0 \left( b + \lambda b ^ { \prime } \right) + 0 \left( c + \lambda c ^ { \prime } \right) = 0 \Rightarrow \lambda = - \frac { a } { a ^ { \prime } }$

Putting the value of λ in (i), the required plane is

$y \left( a ^ { \prime } b - a b ^ { \prime } \right) + z \left( a ^ { \prime } c - a c ^ { \prime } \right) + a ^ { \prime } d - a d ^ { \prime } = 0$

i.e., $\left( a b ^ { \prime } - a ^ { \prime } b \right) y + \left( a c ^ { \prime } - a ^ { \prime } c \right) z + a d ^ { \prime } - a ^ { \prime } d = 0$.