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UPSC 2013 Maths Optional Paper 1 Q5e — Step-by-Step Solution

10 marks · Section B

Curves in space: tangent, normal, binormal · Vector Analysis · asked 2× in 13 yrs · Read the full method →

Question

Show that the curve

lies in a plane.

Technique

Spot a linear combination of $x,y,z$ that simplifies to a constant by examining the component formulas; verify identically in $t$.

Solution

Strategy. Find a fixed plane $\alpha x+\beta y+\gamma z+\delta=0$ that the curve satisfies for all $t$. (Equivalent to showing the torsion vanishes, but spotting the plane is faster.)

Step 1 — Write components

Step 2 — Eliminate $t$

From the $y$ formula: $y-1=\dfrac{1}{t}$.

From the $z$ formula: $z+t=\dfrac{1}{t}$, i.e. $z+x=\dfrac{1}{t}$ (using $x=t$).

Both equal $1/t$, so

Step 3 — Verify the plane equation holds identically

For every $t\ne 0$:

So the curve lies in the plane $\Pi:\;x-y+z+1=0$.

Answer