The largest value of a, for which the perpendicular distance of the plane containing the lines
\( \vec{r} (\hat{i}+\hat{j})+λ(\hat{i}+a\hat{j}−\hat{k})\ and\ \vec{r}=(\hat{i}+\hat{j})+μ(−\hat{i}+\hat{j}−a\hat{k}) \)
from the point (2, 1, 4) is √3, is _____________.
The largest value of a, for which the perpendicular distance of the plane containing the lines
\( \vec{r} (\hat{i}+\hat{j})+λ(\hat{i}+a\hat{j}−\hat{k})\ and\ \vec{r}=(\hat{i}+\hat{j})+μ(−\hat{i}+\hat{j}−a\hat{k}) \)
from the point (2, 1, 4) is √3, is _____________.
Correct Answer: 2
Solution and Explanation
The correct answer is 2
Normal to plane
\(=\left| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & a & -1 \\ -1 & 1 & -a \end{matrix} \right|\)
\(=\hat{i}(1−a^2)−\hat{j}(−a−1)+\hat{k}(1+a)\)
\(=(1−a)\hat{i}+\hat{j}+\hat{k}\)
∴ Plane (1 – a) (x – 1) + (y – 1) + z = 0
Distance from(2,1,4) is \(\sqrt3\)
i.e
\(⇒ |\frac{(1-a)+0+4}{\sqrt{(1-a)^2+1+1}}| = \sqrt3\)
\(⇒ 25+a^2-10a = 3a^2-6a+9\)
\(⇒ 2a^2+4a-16 = 0\)
\(⇒ a^2+2a-8 = 0\)
a=2 or -4
Therefore ,\( a_{max} = 2\)
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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.
