First slide

Planes in 3D

Question

An equation of the plane passing through the line of intersection of the planes $x + y + z = 6$ and $2 x + 3 y + 4 z + 5 = 0$ and passing through $(1, 1, 1)$ is

Moderate

A

$2 x + 3 y + 4 z = 9$
B

$x + y + z = 3$
C

$x + 2 y + 3 z = 6$
D

$20 x + 23 y + 26 z = 69$

Solution

Equation of any plane through the line of intersection of the given planes is
$2 x + 3 y + 4 z + 5 + λ (x + y + z - 6) = 0$
It passes through $(1, 1, 1)$
if $(2 + 3 + 4 + 5) + λ (1 + 1 + 1 - 6) = 0$
$\Rightarrow λ = 14 / 3$ and the required equation is therefore,
$20 x + 23 y + 26 z = 69 .$

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