Maths-
General
Easy

Question

For the ellipse 3x2 + 4y2 – 6x + 8y – 5 = 0

  1. centre is (2, -1)    
  2. eccentricity is fraction numerator 1 over denominator 3 end fraction    
  3. foci are (3, 1) and (-1, 1)    
  4. centre is (1, -1), e = 1 half.    

Hint:

Standard form of ellipse 
x squared over a squared plus y squared over b squared space equals space 1
a n d space e c c e n t r i c i t y left parenthesis e right parenthesis space equals space square root of 1 minus b squared over a squared end root

The correct answer is: centre is (1, -1), e = 1 half.


    Given :
    3 x squared space plus space 4 y squared space – space 6 x space plus space 8 y space – space 5 space equals space 0
    Step 1 Convert in standard form
    rightwards double arrow 3 x squared space plus space 4 y squared space – space 6 x space plus space 8 y space – space 5 space equals space 0

    rightwards double arrow 3 x squared space – space 6 x italic space italic plus italic 3 italic space italic minus italic 3 plus space 4 y squared space space plus space 8 y italic space italic plus italic 4 italic space italic minus italic 4 space – space 5 space equals space 0
rightwards double arrow 3 x squared space – space 6 x italic space italic plus italic 3 italic space plus space 4 y to the power of italic 2 space space plus space 8 y italic space italic plus italic 4 italic space italic minus italic 12 italic space italic equals italic space italic 0
rightwards double arrow 3 left parenthesis x to the power of italic 2 space – space 2 x italic space italic plus italic 1 italic right parenthesis italic space italic plus italic space italic 4 italic left parenthesis y to the power of italic 2 space space plus space 2 y italic space italic plus italic 1 italic right parenthesis italic space italic space italic equals italic space italic 12
D i v i d i n g italic space b y italic space italic 12 italic space o n italic space b o t h italic space s i d e s
    fraction numerator italic rightwards double arrow italic left parenthesis x to the power of italic 2 italic space italic – italic space italic 2 x italic space italic plus italic 1 italic right parenthesis over denominator italic 4 end fraction italic space italic plus italic space fraction numerator italic left parenthesis y to the power of italic 2 italic space italic space italic plus italic space italic 2 y italic space italic plus italic 1 italic right parenthesis italic space italic space over denominator italic 3 end fraction italic equals italic space italic 1
rightwards double arrow italic left parenthesis x italic minus italic 1 italic right parenthesis to the power of italic 2 over italic 4 italic space italic plus italic space italic left parenthesis y italic space italic plus italic space italic 1 italic right parenthesis to the power of italic 2 over italic 3 italic space italic equals italic space italic 1
C o m p a r i n g italic space w i t h italic space s italic tan d a r d italic space e q u a t i o n
rightwards double arrow a to the power of italic 2 italic space italic equals italic space italic 4 italic space a n d italic space b to the power of italic 2 italic space italic equals italic 3
rightwards double arrow C o o r d i n a t e s italic space o f italic space c e n t r e italic space italic equals italic space italic left parenthesis italic 1 italic comma italic minus italic 1 italic right parenthesis
    E c c e n t r i c i t y left parenthesis e right parenthesis space equals space square root of 1 minus b squared over a squared end root
rightwards double arrow e space equals space square root of 1 minus fraction numerator 3 over denominator 4 space end fraction end root space equals space 1 half

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