← 2020 Paper 1

UPSC 2020 Maths Optional Paper 1 Q3c — Step-by-Step Solution

15 marks · Section A

Cone · Analytic Geometry · asked 14× in 13 yrs · Read the full method →

Question

If the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ represents one of a set of three mutually perpendicular generators of the cone $5yz-8zx-3xy=0$, then find the equations of the other two generators.

Technique

The two unknown perpendicular generators = intersection of the cone with the plane through $O$ normal to the known generator; substitute and factor the resulting conic in $(y,z)$.

Solution

The cone is $f(x,y,z)\equiv 5yz-8zx-3xy=0$, passing through the origin.

Step 0 — Existence check

A cone $ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0$ admits sets of three mutually perpendicular generators iff $a+b+c=0$. Here $a=b=c=0$, so $a+b+c=0$ ✓ — such triples exist.

Step 1 — The other two generators lie in the plane through $O$ perpendicular to $(1,2,3)$

If the three generators are mutually perpendicular and one has direction $\mathbf{d}_1=(1,2,3)$, the other two are perpendicular to $\mathbf{d}_1$, hence lie in the plane through the origin with normal $\mathbf{d}_1$:

They are also generators of the cone. So the required two lines are the intersection of the cone with this plane.

Step 2 — Intersect cone with the plane

From the plane, $x=-2y-3z$. Substitute into $5yz-8zx-3xy=0$:

Case A: $y=-z$. Then $x=-2(-z)-3z=-z$, so $(x,y,z)=z(-1,-1,1)$; direction $(1,1,-1)$.

Case B: $y=-4z$. Then $x=-2(-4z)-3z=5z$, so $(x,y,z)=z(5,-4,1)$; direction $(5,-4,1)$.

Step 3 — The two generators

Answer