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Question 581877: Write the following equation in standard form, then identify the type of conic and graph:
64x^2 + 49y^2 +256x - 196y - 2684 = 0
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
You can determine the type of conic by simply examining the given equation.
Both  and  terms exist, so the conic is NOT a parabola.
The coefficients on the and terms are unequal, so the conic is NOT a circle.
The coefficients on the and terms have the same sign, so the conic is NOT a hyperbola.
The conic is therefore an ellipse.
Complete the square on each variable.
Step 1: Constant term to the RHS.
Step 2: Gather like variables:
\ +\ \left(49y^2\ -\ 196y\right)\ =\ 2684)
Step 3: Factor lead coefficient from each of the binomials.
\ +\ 49\left(y^2\ -\ 4y\right)\ =\ 2684)
Step 4: Complete the square on each binomial. Divide the 1st order term coefficient by 2, square the result, add that result inside of the parentheses.
\ +\ 49\left(y^2\ -\ 4y\ +\ 4\right))
Step 5: Take the number added inside of the parentheses times the coefficient outside of the parentheses and add it to the LHS to compensate.
\ +\ 49\left(y^2\ -\ 4y\ +\ 4\right)\ =\ 2684\ +\ 256\ +\ 196\ =\ 3136)
Step 6: Factor the perfect square trinomials in the RHS:
^2\ +\ 49\left(y\ -\ 2\right)^2\ =\ 3136)
Step 7: Divide by the coefficient in the RHS:
^2}{49}\ +\ \frac{(y\ -\ 2)^2}{64}\ =\ 1)
The standard form of the ellipse is:
^2}{a^2}\ +\ \frac{(y\ -\ k)^2}{b^2}\ =\ 1)
where  , the semi-major axis is  and is parallel to the  axis, the semi-minor axis is  , the center is at ) , the vertices are at ) and ) , the minor axis endpoints are at ) and ) , and the foci are at ) and ) where 
or
^2}{b^2}\ +\ \frac{(y\ -\ k)^2}{a^2}\ =\ 1)
where , the semi-major axis is and is parallel to the  axis, the semi-minor axis is , the center is at , the vertices are at and , the minor axis endpoints are at and , and the foci are at ) and ) where
Re-write your equation so that the key values can be seen by inspection. The graph should be obvious from that point:
)^2}{7^2}\ +\ \frac{(y\ -\ 2)^2}{8^2}\ =\ 1)
John
My calculator said it, I believe it, that settles it
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