Maths-
General
Easy

Question

The eccentricity of the curve represented by the equation x2 + 2y2 – 2x + 3y + 2 = 0 is

  1. 0    
  2. fraction numerator 1 over denominator 2 end fraction    
  3. fraction numerator 1 over denominator square root of 2 end fraction    
  4. square root of 2.    

hintHint:

To find eccentricity first convert the equation into standard form i.e.
x squared over a squared space plus space y squared over b squared space equals space 1

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


    To find eccentricity first convert the equation into standard form i.e.
    x squared over a squared space plus space y squared over b squared space equals space 1
    Given :
    rightwards double arrow x squared space plus space 2 y squared space – space 2 x space plus space 3 y space plus space 2 space equals space 0
rightwards double arrow x squared space – space 2 x space plus 1 space minus 1 plus space space 2 left parenthesis y squared plus space 3 over 2 y right parenthesis space plus space 2 space equals space 0
rightwards double arrow left parenthesis x squared space – space 2 x space plus 1 right parenthesis space plus space space 2 left parenthesis y squared plus space 3 over 2 y right parenthesis space plus space 2 space equals space 1
rightwards double arrow left parenthesis x minus 1 right parenthesis squared space plus space 2 left parenthesis y squared space plus space 2 cross times 3 over 2 y space plus 9 over 4 space minus 9 over 4 right parenthesis space plus space 2 space equals space 1
rightwards double arrow left parenthesis x minus 1 right parenthesis squared space plus space 2 left parenthesis y space plus space 3 over 4 right parenthesis squared space minus 9 over 8 space equals space 1
rightwards double arrow left parenthesis x minus 1 right parenthesis squared space plus space 2 left parenthesis y space plus space 3 over 4 right parenthesis squared space space equals space 1 over 8
D i v i d i n g space b y space 1 over 8

rightwards double arrow fraction numerator left parenthesis x minus 1 right parenthesis squared over denominator begin display style 1 over 8 end style end fraction space plus fraction numerator space left parenthesis y space plus space 3 over 4 right parenthesis over denominator begin display style 1 over 16 end style end fraction squared space space equals space 1
    Comparing with standard form
    a squared space equals space 1 over 8 space a n d space b squared space equals space 1 over 16e space equals space square root of 1 space minus space b squared over a squared end root
e space equals space square root of 1 minus space fraction numerator begin display style bevelled 1 over 16 end style over denominator begin display style bevelled 1 over 8 end style end fraction end root space equals square root of 1 minus space 8 over 16 end root space equals space square root of 8 over 16 end root space equals space fraction numerator 1 over denominator square root of 2 end fraction
T h u s comma space T h e space e c c e n t r i c i t y space o f space t h e space c u r v e space r e p r e s e n t e d space b y space t h e space e q u a t i o n space x squared space plus space 2 y squared space – space 2 x space plus space 3 y space plus space 2 space equals space 0 space i s space space fraction numerator 1 over denominator square root of 2 end fraction

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