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UPSC 2023 Maths Optional Paper 1 Q1e — Step-by-Step Solution

10 marks · Section A

Plane · Analytic Geometry · asked 5× in 13 yrs · Read the full method →

Question

A variable plane which is at a constant distance $3p$ from the origin $O$ cuts the axes in the points $A,B,C$ respectively. Show that the locus of the centroid of the tetrahedron $OABC$ is

Technique

Parametrise the variable plane via its unit normal $(l,m,n)$; write down $A,B,C$ as axis intercepts; centroid is the four-vertex average; eliminate $(l,m,n)$ using the unit-norm constraint.

Solution

Step 1 — Parametrise the plane.

A plane at distance $3p$ from $O$ has the form $lx+my+nz=3p$ where $(l,m,n)$ is a unit vector ($l^2+m^2+n^2=1$).

Step 2 — Find $A,B,C$ — intercepts on the axes.

$A$: set $y=z=0$: $lx=3p\Rightarrow x=3p/l$. So $A=(3p/l,\,0,\,0)$. $B=(0,\,3p/m,\,0)$, $C=(0,\,0,\,3p/n)$.

Step 3 — Centroid of tetrahedron $OABC$.

The centroid of a tetrahedron is the average of its four vertices:

Step 4 — Eliminate the direction parameters $l,m,n$.

If $G=(X,Y,Z)$, then $l=\dfrac{3p}{4X},\,m=\dfrac{3p}{4Y},\,n=\dfrac{3p}{4Z}$.

The constraint $l^2+m^2+n^2=1$ becomes

Multiply both sides by $9$:

Answer