The equation of the plane parallel to the lines x − 1 = 2y − 5 = 2z and 3x = 4y − 11 = 3z − 4 and passing through the point (2, 3, 3) is

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x − 4y + 2z + 4 = 0 \text{ Let a, b, c be the direction ratios of the required plane } . \text{ The given line equations can be rewritten as } x - $\frac{1}{1}$ = y - 5/2/1/2 = z - 0/1/2 . . . \left( 1 \right) x - 0/1/3 = y - 11/4/1/4 = z - 4/3/1/3 . . . \left( 2 \right) \text{ Since the required plane is parallel to the lines (1) and } (2), a + b/2 + c/2 = 0 \Rightarrow 2a + b + c = 0 . . . \left( 3 \right) a/3 + b/4 + c/3 = 0 \Rightarrow 4a + 3b + 4c = 0 . . . \left( 4 \right) \text{ Solving (3) and (4) using cross-multiplication method, we get } a/1 = b/- 4 = c/2 = \lambda (\text{say} ) \Rightarrow a = \lambda, b = - 4\lambda, c = 2\lambda \text{ Now, the equation of the plane whose direction ratios are } \lambda, -4\lambda, 2\lambda \text{ and passing through the point (2, 3, 3) is } \lambda \left( x - 2 \right) + \left( - 4\lambda \right)\left( y - 3 \right) + 2\lambda \left( z - 3 \right) = 0 \Rightarrow x - 4y + 2z + 4 = 0

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