Sketching Ellipses: Comparing 0 and 8

[JC3187]

Hi guys,

What is the best way to sketch

0 = x² - 2x + 4y² And
8 = x² - 2x + 4y²

?

How do I sketch these two and how do I know they're both ellipses?
Thank you.

 

[pasmith]

Hi guys,

What is the best way to sketch

0 = x² - 2x + 4y² And
8 = x² - 2x + 4y²

?

How do I sketch these two and how do I know they're both ellipses?
Thank you.

Complete the square in $x$: $x^2 - 2ax = (x - a)^2 - a^2$.

 

[JC3187]

Then what do i do?

 

[CAF123]

Then what do i do?

What is the general equation for an ellipse in Cartesian coordinates? Can you see why completing the square helps?

 

[HallsofIvy]

$$(x- 1)^2+ y^2= x^2- 2x+ 1+ y^2$$
Do you see how that is connected to $$x^2+ 2x+ y^2$$?