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UPSC 2015 Maths Optional Paper 1 Q2d — Step-by-Step Solution

13 marks · Section A

Cone · Analytic Geometry · asked 15× in 14 yrs · Read the full method →

Question

If $6x=3y=2z$ represents one of the three mutually perpendicular generators of the cone $5yz-8zx-3xy=0$, then obtain the equations of the other two generators.

Technique

Use the perpendicularity-of-three-mutually-perpendicular-generators standard recipe: one constraint from cone equation + one from perpendicularity to the given generator gives a quadratic in $m/n$; the two roots are the directions of the other two generators; their mutual perpendicularity is then automatic (standard result for the three principal generators).

Solution

Setup. The given generator has direction ratios satisfying $6x=3y=2z$, i.e. $\dfrac{x}{1}=\dfrac{y}{2}=\dfrac{z}{3}$. Direction $\vec\ell_1=(1,2,3)$.

A line through the origin $\vec r=t(l,m,n)$ lies on the cone $5yz-8zx-3xy=0$ iff its direction $(l,m,n)$ satisfies

Check $(1,2,3)$: $5(6)-8(3)-3(2)=30-24-6=0$ ✓.

Strategy. We seek $\vec\ell_2=(l_2,m_2,n_2)$ and $\vec\ell_3=(l_3,m_3,n_3)$, both on the cone, both perpendicular to $\vec\ell_1=(1,2,3)$, and perpendicular to each other.

Step 1 — Perpendicularity to $\vec\ell_1$

Step 2 — Substitute into the cone equation $(\star)$

$5mn-8nl-3lm=0$.

$l=-2m-3n$ gives $nl=-2mn-3n^2$ and $lm=-2m^2-3mn$.

So either $m=-n$ or $m=-4n$.

Step 3 — Case $m=-n$

From (1): $l=-2(-n)-3n=2n-3n=-n$. Direction $(l,m,n)=(-n,-n,n)=n(-1,-1,1)$, i.e. $\vec\ell_2=(-1,-1,1)$ (or equivalently $(1,1,-1)$).

Step 4 — Case $m=-4n$

From (1): $l=-2(-4n)-3n=8n-3n=5n$. Direction $(l,m,n)=(5n,-4n,n)=n(5,-4,1)$, i.e. $\vec\ell_3=(5,-4,1)$.

Step 5 — Verify all three are mutually perpendicular

• $\vec\ell_1\cdot\vec\ell_2=(1,2,3)\cdot(-1,-1,1)=-1-2+3=0$ ✓.
• $\vec\ell_1\cdot\vec\ell_3=(1,2,3)\cdot(5,-4,1)=5-8+3=0$ ✓.
• $\vec\ell_2\cdot\vec\ell_3=(-1,-1,1)\cdot(5,-4,1)=-5+4+1=0$ ✓.

Verify each lies on the cone $(\star)$:

• $\vec\ell_2=(-1,-1,1)$: $5(-1)(1)-8(1)(-1)-3(-1)(-1)=-5+8-3=0$ ✓.
• $\vec\ell_3=(5,-4,1)$: $5(-4)(1)-8(1)(5)-3(5)(-4)=-20-40+60=0$ ✓.

Conclusion

The other two generators are the lines through the origin with direction ratios $(-1,-1,1)$ and $(5,-4,1)$:

Answer