Plane Through Line Of Intersection

Question

A family of planes passes through the line of intersection of x + y + z = 1 and 2x + 3y - z = 4 in space.

RankGuru · Accepted Answer

The governing concept is the pencil-of-planes idea: any plane through the line of intersection of P_1 = 0 and P_2 = 0 can be written as P_1 + \lambda P_2 = 0 for some scalar \lambda. This linear-combination representation is a powerful and frequently examined JEE Advanced construction. Writing P_1 = x + y + z - 1 and P_2 = 2x + 3y - z - 4, the family becomes (x + y + z - 1) + \lambda(2x + 3y - z - 4) = 0. Any point on both original planes makes P_1 and P_2 zero, hence satisfies this combination for every \lambda, so the whole line of intersection lies in each member. Option two drops the constant terms, so its members no longer pass through the correct line. Option three is a product representing two separate planes, not a single plane. Option four mismatches the structure and the constant. This applies the family-of-planes theorem. Plausibility check: substituting any point common to P_1 = 0 and P_2 = 0 yields 0 + \lambda \cdot 0 = 0, confirming every such plane contains the intersection line.

Plane Through Line Of Intersection

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