devarshi-dt-logo

Question:

Let S = {(x,y) ∈ R³: y²/(1+r) - x²/(1-r) = 1, where r ≠ ±1. Then S represents:

A hyperbola whose eccentricity is 2√(r+1), where 0 < r < 1.

An ellipse whose eccentricity is √(r+1), where r > 1.

A hyperbola whose eccentricity is 2√(1-r), when 0 < r < 1.

An ellipse whose eccentricity is √(2r+1), when r > 1

Solution:

y²/(1+r) - x²/(1-r) = 1
For r > 1, y²/(1+r) + x²/(r-1) = 1
e = √(1 - ((r-1)/(1+r))) = √((1+r) - (r-1))/(1+r) = √(2r+1) = √(2r+1)