If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky – 5z = 1 and 3kx – ky + z = 5, k < 3 and intercepts a unit length on positive x-axis, then the intercept made by the plane P on the y-axis is
If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky – 5z = 1 and 3kx – ky + z = 5, k < 3 and intercepts a unit length on positive x-axis, then the intercept made by the plane P on the y-axis is
\(\frac{1}{11}\)
\(\frac{5}{11}\)
- 6
- 7
The Correct Option is D
Solution and Explanation
The correct answer is (D):
P1: 2x+ky-5z = 1
P2 : 3kx – ky + z = 5
∵ P1⊥P2 ⇒ 6k-k2+5 = 0
⇒ k = 1,5
∵ k<3
∴ k = 1
P1 : 2x+y-5z = 1
P2 : (2x+y-5z-1)+λ(3x-y+z-5) = 0
Positive x-axis intercept = 1
\(⇒ \frac{1+5λ}{2+3λ} = 1\)
⇒ λ = \(\frac{1}{2}\)
∵ P : 7x+y-4z = 7
y intercept = 7.
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Concepts Used:
Angle between a Line and a Plane
A line is one example of a one-dimensional figure, which has length but no width. A line is made up of a set of points that is stretched in opposite directions infinitely.
Similarly, when an infinite number of points expanded infinitely in either direction to form a flat surface, it is known as a plane. A set of lines when arranged close by to each other a plane is obtained. A plane is one example of a two-dimensional geometric figure that can be measured in terms of length and width.
The line which is adjacent to the plane is the complement of the angle between and the normal of the plane is called the angle between a line and a plain.