Answer:
Equation of the Ellipse
[tex]\frac{(x+1)^{2} }{8} + \frac{(y-3)^{2} }{4} = 1[/tex]
Step-by-step explanation:
Step(i):-
Given that the equation
x² + 2 x + 2y² - 12 y +11 = 0
⇒ x² + 2 x + 1 - 1 + 2(y² - 6 y )+ 11 = 0
x² + 2 x + 1 - 1 + 2(y² - 2(3) y+9-9 )+ 11 = 0
⇒ x² + 2 x + 1 - 1 + 2(y² - 2(3 y ) + 3²- 3² ) + 11 = 0
By using (a +b)² = a² + 2 a b + b²
(a -b)² = a² - 2 a b + b²
Step(ii):-
x² + 2 x + 1 - 1 + 2(y² - 2(3 y ) + 3²- 3² ) + 11 = 0
⇒ ( x+1)² +2( y-3 )² - 1 - 2(9) +11 =0
⇒ ( x+1)² +2( y-3 )² - 8 =0
( x+1)² +2( y-3 )² = 8
Dividing '8' on both sides , we get
[tex]\frac{(x+1)^{2} }{8} + \frac{2(y-3)^{2} }{8} = 1[/tex]
[tex]\frac{(x+1)^{2} }{8} + \frac{(y-3)^{2} }{4} = 1[/tex]
This equation represents the Ellipse
