Question

If e₁, e₂ be the eccentricities of two conics S₁ and S₂ and if \(e_1^2 + e_2^2 = 3\) then both S₁ and S₂ can be
1. ellipse
2. parabolas
3. hyperbolas
4. circles

Answer 1

Correct Answer - Option 3 : hyperbolas

Concept:

For an Ellipse the eccentricity is less than 1

For a Hyperbola the eccentricity is greater than 1

Calculation:

\(e_1^2 + e_2^2 = 3\)

e₁​ and e₂​ are eccentricities of same type of conic.

Let e₁ ​ = e₂

⇒ 2e₁² = 3

⇒ e12 = 1.5 

⇒ e1 = √1.5 > 1

∴ Both conics S1 and S2 is Hyperbolas