Question:

In $R^3$, Let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P_1 : x + 2y - z + 1 = 0$ and $P_2 : 2x - y + z - 1 = 0$. Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ to the plane $P_1$. Which of the following points lie(s) on $M$ ?

Updated On: Jun 14, 2022
  • $\left(0, -\frac{5}{6}, -\frac{2}{3}\right)$
  • $\left( -\frac{1}{6}, -\frac{1}{3}, \frac{1}{6}\right)$
  • $\left( -\frac{5}{6}, 0, \frac{1}{6}\right)$
  • $\left( -\frac{1}{3}, 0, \frac{2}{3}\right)$
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is B

Solution and Explanation

$P_{1} : x + 2y - z + 1 = 0$
$\& \,P_{2} : 2x - y + z - 1 = 0$
Direction Ratios of common line $\left(1, -3, -5\right) ? \hat{i}-3\hat{j}-5\hat{k}$
$L : \frac{x}{1} = \frac{y}{-3} = \frac{z}{-5} = t$
Let M$\left(a, ?, ?\right)$ is feet of perpendicular from $\left(t, -3t, - 5t\right)$ on $P_{1}$
$\frac{\alpha-t}{1} = \frac{\beta+3t}{2} = \frac{\gamma+5t}{-1} = -\left(\frac{t-6t+5t+1}{6}\right)$
$\alpha = t-\frac{1}{6}\quad\beta = -3t -\frac{1}{3}\quad\gamma = -5t + \frac{1}{6}$
Only option $\left(A\right) \& \left(B\right)$ satisfies.
Was this answer helpful?
0
0

Top Questions on Three Dimensional Geometry

View More Questions

Questions Asked in JEE Advanced exam

View More Questions

Concepts Used:

Three Dimensional Geometry

Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.

Direction Cosines and Direction Ratios of Line:

Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.

Three Dimensional Geometry