First slide

Planes in 3D

Question

The equation of the plane through the line of intersection of the planes ax + by + cz + d = 0 and $dx + b^{'} y + c^{'} z + d^{'} = 0$ and parallel to the line y=0 and z=0 is

Moderate

A

$({ab}^{'} - a^{'} b) x + ({bc}^{'} - b^{'} c) y + ({ad}^{'} - a^{'} d) = 0$
B

$(a b^{'} - a^{'} b) x + (b c^{'} - b^{'} c) y + (a d^{'} - a^{'} d) z = 0$
C

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

none of these

Solution

The equation of a plane through the line of intersection of the planes ax + by + cz + d = 0 and $a^{'} x + b^{'} y + c^{'} z + d^{'} = 0$ is
$\begin{array}{l} (ax + by + cz + d) + λ (a^{'} x + b^{'} y + c^{'} z + d^{'}) = 0 \\ or x (a + {λa}^{'}) + y (b + {λb}^{'}) + z (c + {λc}^{'}) + d + {λd}^{'} = 0 - - - - (i) \end{array}$
This is parallel to the x-axis, i.e., y=0, z = 0.
Therefore, $1 (a + {λa}^{'}) + 0 (b + {λb}^{'}) + 0 (c + {λc}^{'}) = 0$
or, $λ = - \frac{a}{a^{'}}$
Putting the value of $λ$ in (i), the required plane is
$\begin{aligned} y (a^{'} b - {ab}^{'}) + z (a^{'} c - {ac}^{'}) + a^{'} d - {ad}^{'} = 0 \\ or ({ab}^{'} - a^{'} b) y + ({ac}^{'} - a^{'} c) z + {ad}^{'} - a^{'} d = 0 \end{aligned}$

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