Question

Assertion :The shortest distance between the skew lines $\frac{x-1}{2}=\frac{x-1}{-1}=\frac{z-0}{1}$ and $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+1}{2}$ is $\frac{10}{\sqrt{59}}$ Reason: Two lines are skew if there exists no plane passing through them

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is B Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion

Let $l,m,n$ be the DC's of the line of the common perpendicular (or SD) to the two given lines.

$\therefore 2l-m+n=0$ & $3l-5m+2n=0$

On solving for $l,m,n$, we have

$\frac{l}{3}=\frac{m}{-1}=\frac{n}{-7}=\frac{\sqrt{l^2+m^2+n^2}}{\sqrt{3^2+{(-1)}^2+7^2}}=\frac{1}{\sqrt{59}}$

$\Rightarrow$ DC's of SD line are $\frac{3}{\sqrt{59}},-\frac{1}{\sqrt{59}},\frac{-7}{\sqrt{59}}$

Also the given lines passes through the points $(1,1,0)$ and $(2,1,-1)$.

$\therefore$ Shortest distance =$l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1)$

$=\frac{3}{\sqrt{59}}(2-1)+\frac{(-1)}{\sqrt{59}}(1-1)+\frac{(-7)}{\sqrt{59}}(-1-0)=\frac{10}{\sqrt{59}}$

$\therefore$ Assertion is true

Also the non parallel,non intersecting & non coplanar lines are said to be skew lines.

Hence, Assertion (A) & Reason (R) both are true but reason (R) is not the correct explanation for Assertion (A).