Suppose, the line (x - 2)/α = (y - 2)/(- 5) = (z + 2)/2 lies on the plane x + 3y - 2z + β = 0. Then (α + β ) is equal to _______.

Solution

Maths sir · 4 steps

A line lies in a plane only if it runs parallel to the plane and one of its points sits on it.

1. Read off the line. The line passes through (2, 2, -2) and has direction ratios (α, -5, 2). The plane's normal is (1, 3, -2).
2. Parallel condition. A line lying in the plane is perpendicular to the normal, so the dot product of its direction with the normal is zero.

   α(1) + (-5)(3) + (2)(-2) = 0 ⇒ α - 19 = 0 ⇒ α = 19

3. Point on plane. The point (2, 2, -2) must satisfy the plane: 2 + 3(2) - 2(-2) + β = 0, so 12 + β = 0 and β = -12.
4. So. α + β = 19 - 12 = 7.

The common slip: 15. Reading the point as (2, 2, 2) instead of (2, 2, -2) from z + 2 gives β = -4 and a sum of 15.

The idea to keep. For a line to lie in a plane, check two things: direction dotted with the normal is zero, and any one point of the line satisfies the plane. One gives a slope-type unknown, the other gives the constant.

Answer from the NTA final answer key. Solution by PrepWiser.