Let S = x , y ∈ R 2 : y 2 1 + r - x 2 1 - r = 1 , where r ≠ ± 1 . Then S represents:

Let S=x,y∈R2:y21+r-x21-r=1, where r≠±1. Then S represents:
  1. An ellipse whose eccentricity is 1r+1, when r>1.
  2. A hyperbola whose eccentricity is 2r+1, when 0<r<1.
  3. An ellipse whose eccentricity is 2r+1, when r>1
  4. A hyperbola whose eccentricity is 21-r, when 0<r<1

Solution

Given y21+r-x21-r=1

If r∈0,1 then this curve is a hyperbola whose eccentricity is 1+1-r1+r=2r+1. 

If r>1 , then this curve is an ellipse whose
eccentricity is 1-r-1r+1=2r+1. 

Asked in: JEE Main 2019 (10 Jan Shift 2)

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