The sum of all values of , for which the shortest distance between the lines x + 1 = y - 2 - 1 = z - 4 - and x = y - 1 2 = z - 1 2 is 2 , is [2026]

Question

The sum of all values of $α$ , for which the shortest distance between the lines

$\frac{x + 1}{α} = \frac{y - 2}{- 1} = \frac{z - 4}{- α} and \frac{x}{α} = \frac{y - 1}{2} = \frac{z - 1}{2 α}$ is $\sqrt{2}$ , is [2026]

Smart Achievers · Accepted Answer

(2)

$\sqrt{2} =$ $\frac{| \begin{matrix} - 1 & 1 & 3 \\ α & - 1 & - α \\ α & 2 & 2 α \end{matrix} |}{| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ α & - 1 & - α \\ α & 2 & 2 α \end{matrix} |}$

$\sqrt{2} = \frac{- 1 (- 2 α + 2 α) - 1 (2 α^{2} + α^{2}) + 3 (2 α + α)}{| \hat{i} (- 2 α + 2 α) - \hat{j} (2 α^{2} + α^{2}) + \hat{k} (2 α + α) |}$

$\sqrt{2} = \frac{- 3 α^{2} + 9 α}{\sqrt{9 α^{4} + 9 α^{2}}}$

$\sqrt{2} = \frac{- α + 3}{\sqrt{α^{2} + 1}}$

$\Rightarrow 2 α^{2} + 2 = α^{2} + 9 - 6 α$

$α^{2} + 6 α - 7 = 0$

$(α + 7) (α - 1) = 0$

$α = - 7, 1$

$sum = - 7 + 1 = - 6$

option (2)

Solution

Q.
The sum of all values of $α$ , for which the shortest distance between the lines

$\frac{x + 1}{α} = \frac{y - 2}{- 1} = \frac{z - 4}{- α} and \frac{x}{α} = \frac{y - 1}{2} = \frac{z - 1}{2 α}$ is $\sqrt{2}$ , is [2026]

1 8

2 - 6

3 6

4 - 8

Want Us to Email you About Special Offers & Updates?

Solution

Q. The sum of all values of α, for which the shortest distance between the lines x+1α=y-2-1=z-4-α and xα=y-12=z-12α is 2, is [2026]

1 8

2 - 6

3 6

4 - 8

Ans. (2) 2=|-113α-1-αα22α||i^j^k^α-1-αα22α| 2=-1(-2α+2α)-1(2α2+α2)+3(2α+α)|i^(-2α+2α)-j^(2α2+α2)+k^(2α+α)| 2=-3α2+9α9α4+9α2 2=-α+3α2+1 ⇒2α2+2=α2+9-6α α2+6α-7=0 (α+7)(α-1)=0 α=-7, 1 sum=-7+1=-6 option (2)

Want Us to Email you About Special Offers & Updates?

Q.
The sum of all values of $α$ , for which the shortest distance between the lines

$\frac{x + 1}{α} = \frac{y - 2}{- 1} = \frac{z - 4}{- α} and \frac{x}{α} = \frac{y - 1}{2} = \frac{z - 1}{2 α}$ is $\sqrt{2}$ , is [2026]