Find the equation of the plane which passes through $a_{1} x + b_{1} x + c_{1} z + d_{1} = 0$ , $a_{2} x + b_{2} x + c_{2} z + d_{2} = 0$ and which is parallel to the line $\frac{x - α}{l} = \frac{y - β}{m} = \frac{z - γ}{n}$

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Solution

The equation of plane passing through $a_{1} x + b_{1} y + c_{1} z + d_{1} = 0, a_{2} x + b_{2} y + c_{2} z + d_{2} = 0$ will be $(a_{1} x + b_{1} y + c_{1} z + d_{1}) + λ (a_{2} x + b_{2} y + c_{2} z + d_{2}) = 0$
$⟹ (a_{1} + λ a_{2}) x + (b_{1} + λ b_{2}) y + (c_{1} + λ c_{2}) z + (d_{1} + λ d_{2}) = 0$
Since this plane is parallel to $\frac{x - α}{l} = \frac{y - β}{m} = \frac{z - γ}{n}$
$(a_{1} + λ a_{2}) l + (b_{1} + λ b_{2}) m + (c_{1} + λ c_{2}) n = 0 ⟹ λ = - \frac{a_{1} l + b_{1} m + c_{1} n}{a_{2} l + b_{2} m + c_{2} n}$
The plane equation is $(a_{1} x + b_{1} y + c_{1} z + d_{1}) - \frac{(a_{1} l + b_{1} m + c_{1} n)}{(a_{2} l + b_{2} m + c_{2} n)} (a_{2} x + b_{2} y + c_{2} z + d_{2}) = 0$

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a_{1} x + b_{1} y + c_{1} z + d_{1} = 0

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