Question

The largest value of a, for which the perpendicular distance of the plane containing the lines ${\displaystyle \overrightarrow{\text{r}}=(\hat{\text{i}}+\hat{\text{j}})+\lambda (\hat{\text{i}}+\text{a}\hat{\text{j}}-\hat{\text{k}})}$ and ${\displaystyle \overrightarrow{\text{r}}=(\hat{\text{i}}+\hat{\text{j}})+\mu (-\hat{\text{i}}+\hat{\text{j}}-\text{a}\hat{\text{k}})}$ from the point (2, 1, 4) is ${\displaystyle \sqrt{3}}$, is ______.

Options

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Answer 1

The largest value of a, for which the perpendicular distance of the plane containing the lines ${\displaystyle \overrightarrow{\text{r}}=(\hat{\text{i}}+\hat{\text{j}})+\lambda (\hat{\text{i}}+\text{a}\hat{\text{j}}-\hat{\text{k}})}$ and ${\displaystyle \overrightarrow{\text{r}}=(\hat{\text{i}}+\hat{\text{j}})+\mu (-\hat{\text{i}}+\hat{\text{j}}-\text{a}\hat{\text{k}})}$  from the point (2, 1, 4) is ${\displaystyle \sqrt{3}}$, is 2.

Explanation:

Given vectors are

${\displaystyle \overrightarrow{\text{r}}=(\hat{\text{i}}+\hat{\text{j}})+\lambda (\hat{\text{i}}+\text{a}\hat{\text{j}}-\hat{\text{k}})}$ and ${\displaystyle \overrightarrow{\text{r}}=(\hat{\text{i}}+\hat{\text{j}})+\mu (-\hat{\text{i}}+\hat{\text{j}}-\text{a}\hat{\text{k}})}$ 

D.R’s of the plane containing these lines is

${\displaystyle \left|\begin{array}{ccc}\hat{\text{i}}& \hat{\text{j}}& \hat{\text{k}}\\ 1& \text{a}& -1\\ -1& 1& -\text{a}\end{array}\right|=\hat{\text{i}}{(1-\text{a})}^2-\hat{\text{j}}(-\text{a}-1)+\hat{\text{k}}(1+\text{a})}$

${\displaystyle \overrightarrow{\text{n}}=(1-\text{a})\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}}$

One point in the plane : (1, 1, 0)

∴ equation of the plane is (1 – a)(x – 1) + (y – 1) + (z – 0) = 0

${\displaystyle \Rightarrow }$ (1 – a)x + y + z + a – 2 = 0

So,

D = ${\displaystyle \frac{|(1-\text{a})2+1+4+\text{a}-2|}{\sqrt{{(1-\text{a})}^2+1+1}}}$

${\displaystyle \Rightarrow }$ |5 – a| = ${\displaystyle \sqrt{3}.\sqrt{{\text{a}}^2-2\text{a}+3}}$

${\displaystyle \Rightarrow }$ a² + 2a – 8 = 0

${\displaystyle \Rightarrow }$ a = 2, – 4

Therefore, largest value of a = 2