SOLUTION: What type of graph does this equation represent? x^2 + 4y^2

SOLUTION: What type of graph does this equation represent? x^2 + 4y^2 - 2y = 8 THANKS!

Question 308967: What type of graph does this equation represent?
x^2 + 4y^2 - 2y = 8

THANKS!
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation.

Factor a 4 from the last two terms on the left side.

Now take half of to get . Square it to get . Add AND subtract this value inside the parenthesis.

Group up the first three terms inside the parenthesis.

Factor that inner group to get

Distribute.

Multiply

Reduce.

Add to both sides.

Combine like terms.

Multiply EVERY term (outside the parenthesis) by the LCD 4 to clear out the fraction.

Divide both sides by 33 (to make the right side equal to 1).

Break up the fraction.

Rearrange the terms in the fractions.

Rewrite as and as

Finally, write as

It might be hard to notice, but the last equation above is in the form which is an ellipse.

So is an ellipse

This means that is also an ellipse (since the two equations are equivalent)

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This is a lot of work to determine the conic. An alternative is to use the following rule:

For the general conic

If , then the given conic above is an ellipse
Furthermore, if , and , then the conic is also a circle

If , then the given conic above is a parabola

If , then the given conic above is a hyperbola

In our case, we have the conic which means that , , , , , and (subtract this from both sides). Plug these values into to get which is indeed less than 0 meaning that it is an ellipse.

Personally, I recommend doing it the long way for a while at first so you get a feel of what you are doing. Once you understand what's going on, take the shortcut.
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