SOLUTION: Write the following equation in standard form, then identify the type of conic and graph:
64x^2 + 49y^2 +256x - 196y - 2684 = 0
Algebra.Com
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
RELATED QUESTIONS
Write the following equation in standard form, then identify the type of conic and... (answered by lwsshak3)
Write the following equation in standard form, then identify the type of conic
9x^2 -... (answered by stanbon)
Identify the type of conic and write the equation in standard form:
x^2+y^2-6x+4y+3=0
(answered by lwsshak3)
Classify the conic section and write its equation in standard form.... (answered by Big Poop)
3x^2-5x+3y^2-8=0
Write the equation in standard form. Identify which type of conic... (answered by lwsshak3)
Identify the conic section, write each in standard form, and graph:... (answered by stanbon)
Identify the conic section, re-write the equation in standard form, list the main... (answered by lwsshak3)
Identify the conic(cirlce, ellipse, hyperbola, ellipse. Write each of the following in... (answered by jim_thompson5910)
put the equation 16x^2+y^2+64x-2y+67=0 into conic standard... (answered by Edwin McCravy,MathLover1,ikleyn)
