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UPSC 2014 Maths Optional Paper 1 Q4b — Step-by-Step Solution

15 marks · Section A

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

Question

Show that the lines drawn from the origin parallel to the normals to the central conicoid $ax^{2}+by^{2}+cz^{2}=1$, at its points of intersection with the plane $lx+my+nz=p$ generate the cone

Technique

Parametrise the lines; use the two given surface conditions to eliminate the foot-point coordinates.

Solution

Strategy. At a point $(x_0,y_0,z_0)$ on the conicoid, the normal direction is given by the gradient. A line from the origin parallel to this normal is $(x,y,z)=t(ax_0,by_0,cz_0)$. Eliminate the parameters $(x_0,y_0,z_0,t)$ using the two given constraints.

Step 1 — Normal at a point on the conicoid

$F(x,y,z)=ax^{2}+by^{2}+cz^{2}-1$, so $\nabla F=(2ax,2by,2cz)$. At $(x_0,y_0,z_0)$, normal direction is $(ax_0,by_0,cz_0)$.

Step 2 — Parametrise line from origin

Step 3 — Apply the two constraints

Conicoid constraint $ax_0^{2}+by_0^{2}+cz_0^{2}=1$:

hence

Plane constraint $lx_0+my_0+nz_0=p$:

hence

Step 4 — Eliminate $t$

Square (B):

Equate with (A):

Multiply by $p^{2}$:

Answer