Hyperbola 1 Question 5

6. Let S=(x,y)R2:y21+rx21r=1,

where r±1. Then, S represents (2019 Main, 10 Jan II)

(a) a hyperbola whose eccentricity is 21r, when 0<r<1.

(b) a hyperbola whose eccentricity is 2r+1, when 0<r<1. (c) an ellipse whose eccentricity is 2r+1, when r>1.

(d) an ellipse whose eccentricity is 1r+1, when r>1.

Show Answer

Answer:

Correct Answer: 6. (c)

Solution:

  1. Given, S=(x,y)R2 : y21+rx21r=1

=(x,y)R2:y21+r+x2r1=1

For r>1,y21+r+x2r1=1, represents a vertical ellipse.

[ for r>1,r1<r+1 and r1>0]

Now, eccentricity (e)=1r1r+1

 For x2a2+y2b2=1,a<b,e=1a2b2=(r+1)(r1)r+1=2r+1



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