Maths-
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Question

An ellipse has OB as a semi – minor axis, F, F' are its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is

  1. fraction numerator 1 over denominator square root of 2 end fraction    
  2. fraction numerator 1 over denominator 2 end fraction    
  3. square root of fraction numerator 3 over denominator 2 end fraction end root    
  4. none of these    

Hint:

Eccentricity of ellipse formula is given by
e space equals space square root of 1 minus b squared over a squared end root

The correct answer is: fraction numerator 1 over denominator square root of 2 end fraction



    G i v e n comma space
F space a n d space F space space apostrophe space a r e space f o c i space o f space a n space e l l i p s e comma space w h o s e space c o o r d i n a t e s space a r e space left parenthesis a e comma 0 right parenthesis space a n d space left parenthesis negative a e comma 0 right parenthesis space r e s p e c t i v e l y space
a n d space c o o r d i n a t e s space o f space B space a r e space left parenthesis 0 comma b right parenthesis.
    rightwards double arrow S l o p e space o f space B F apostrophe space space equals space fraction numerator b over denominator negative a e end fraction

rightwards double arrow S l o p e space o f space B F space space equals space fraction numerator b over denominator a e end fraction
    rightwards double arrow S i n c e comma space angle F B F space space apostrophe space space space space equals 90 degree
      m1m2 = -1
    rightwards double arrow negative fraction numerator b over denominator a e end fraction fraction numerator b over denominator a e end fraction space equals space minus 1
rightwards double arrow b squared space equals space a squared e squared
rightwards double arrow e squared space equals space b squared over a squared space minus negative negative negative negative negative space left parenthesis 1 right parenthesis
    Eccentricity of ellipse formula is given by
    e space equals space square root of 1 minus b squared over a squared end root
    rightwards double arrow e squared space equals space 1 minus b squared over a squared
    From equation (1)
    rightwards double arrow e squared space equals space 1 space minus space e squared
rightwards double arrow 2 e squared space equals space 1
rightwards double arrow e squared space equals 1 half
rightwards double arrow e space equals space fraction numerator 1 over denominator square root of 2 end fraction
    Thus,  the eccentricity of the ellipse is fraction numerator 1 over denominator square root of 2 end fraction

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