← 2014 Paper 1

UPSC 2014 Maths Optional Paper 1 Q4c — Step-by-Step Solution

15 marks · Section A

Hyperboloid of one sheet · Analytic Geometry · asked 2× in 13 yrs · Read the full method →

Question

Find the equations of the two generating lines through any point $(a\cos\theta,\,b\sin\theta,\,0)$ of the principal elliptic section $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\;z=0$, of the hyperboloid by the plane $z=0$.

Technique

Standard two-family factorisation of hyperboloid; substitute the point to find the parameter $\lambda$ or $\mu$ of each generator; compute direction.

Solution

Strategy. The hyperboloid of one sheet $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-\dfrac{z^{2}}{c^{2}}=1$ has two ruling families. Through each point of the principal section, one line of each family passes. Identify the parameters of each line; then write line equations.

Step 1 — Ruling families

Factor the hyperboloid:

Family $\Lambda$ (parameter $\lambda$):

Family $M$ (parameter $\mu$):

Step 2 — Determine $\lambda,\mu$ at $P=(a\cos\theta,b\sin\theta,0)$

At $P$: $x/a=\cos\theta$, $y/b=\sin\theta$, $z/c=0$.

Substitute into family-$\Lambda$ first equation: $\cos\theta=\lambda(1-\sin\theta)\Rightarrow\lambda=\dfrac{\cos\theta}{1-\sin\theta}$.

Substitute into family-$M$ first equation: $\cos\theta=\mu(1+\sin\theta)\Rightarrow\mu=\dfrac{\cos\theta}{1+\sin\theta}$.

(Verify the second equation in each family holds — it does, by the identity $\cos^{2}\theta=(1-\sin\theta)(1+\sin\theta)$.)

Step 3 — Direction vectors of the two generators

For a line of family $\Lambda$, parameter $\lambda$, the direction vector (proportional to $\partial/\partial y$) works out to be (after computation):

Substituting $\lambda=\cos\theta/(1-\sin\theta)$ and simplifying (using $1-\lambda^{2}=-2\sin\theta/(1-\sin\theta)$ and $1+\lambda^{2}=2/(1-\sin\theta)$):

Similarly for family $M$:

Step 4 — Line equations through $P$

Generator from family $\Lambda$:

Generator from family $M$:

Answer