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in Three Dimensional Geometry by (28.9k points)
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Find the equation of the plane passing through the intersection of the planes x + y + 4z + 5 = 0 and 2x – y + 3z + 6 = 0 and contains the point (1, 0, 0).

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The equation of the planes passing through the intersection of the planes

x + y + 4z + 5 = 0 and 2x – y + 3z + 6 = 0 is

x + y + 4z + 5 + k(2x – y + 3z + 6) = 0 ____(1)

Since (1) pass through (1, 0, 0)

⇒ 1 + 0 + 0 + 5 + k(2 – 0 + 0 + 6) = 0

⇒ 6 + 8k = 0 ⇒ k = –\(\frac{3}{4}\); Then (1)

⇒ x + y + z + 5 + \(\frac{3}{4}\)(2x – y + 3z + 6) = 0

⇒ 4x + 4y + 16z + 20 – 6x + 3y – 9z – 18 = 0

⇒ 2x – 7y – 7z = 2.

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