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UPSC 2018 Maths Optional Paper 1 Q2d — Step-by-Step Solution

12 marks · Section A

Shortest distance between two skew lines · Analytic Geometry · asked 4× in 13 yrs · Read the full method →

Question

Find the shortest distance between the lines

and the $z$-axis.

Technique

Skew-line distance $\dfrac{|(A-O)\cdot(\vec u\times\vec v)|}{|\vec u\times\vec v|}$ with $\vec u=\vec n_1\times\vec n_2$ and $\vec v=\hat k$; the cross product collapses to $(q,-p,0)$.

Solution

Step 1 — Direction of the given line $L$

$L$ is the intersection of the two planes, so its direction is the cross product of their normals $\vec n_1=(a_1,b_1,c_1),\ \vec n_2=(a_2,b_2,c_2)$:

Step 2 — Skew-lines distance formula

The $z$-axis passes through $O=(0,0,0)$ with direction $\vec v=(0,0,1)$. For two lines through $A$ (on $L$) and $O$ with directions $\vec u,\vec v$, the shortest distance is

Now

Step 3 — A point $A$ on $L$ and the final formula

Take the point of $L$ in the plane $z=0$: solve $a_1x+b_1y+d_1=0,\ a_2x+b_2y+d_2=0$, giving (Cramer)

(The denominator is $r=a_1b_2-a_2b_1$.) Then $(A-O)\cdot(q,-p,0)=qA_x-pA_y$, so

Answer