← 2019 Paper 1

UPSC 2019 Maths Optional Paper 1 Q2c-i — Step-by-Step Solution

10 marks · Section A

Sphere · Analytic Geometry · asked 17× in 13 yrs · Read the full method →

Question

The plane $x+2y+3z=12$ cuts the axes of coordinates in $A,B,C$. Find the equations of the circle circumscribing the triangle $ABC$.

Technique

Circle through three points in space $=$ (sphere through the three points + origin) $\cap$ (their plane). Find intercepts, fit the origin-sphere, present both equations.

Solution

Step 1 — The intercept points

Setting two coordinates to zero in $x+2y+3z=12$:

The circumscribing circle of triangle $ABC$ is the intersection of the given plane with a sphere through $A,B,C$. A convenient such sphere is the one through $A,B,C$ and the origin $O$ (four points determine a sphere; this circle then lies on it and on the plane).

Step 2 — Sphere through $O,A,B,C$

A sphere through the origin has the form

Impose passage through $A,B,C$:

Hence the sphere is

Step 3 — The circle as plane $\cap$ sphere

The required circle is the pair of simultaneous equations

Answer