Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
-

17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0

\theta = 26.6 ^ { \circ }

\frac { x ^ { \prime 2 } } { 4 } + \frac { y ^ { \prime 2 } } { 16 } = 1

ellipse
center at

( 0,0 )

major axis is

y ^ { \prime } -

axis
vertices at

( 0 , \pm 4 )

\theta = 63.4 ^ { \circ }

\frac { \left( x ^ { \prime } - \frac { 2 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } + \frac { 4 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 4 } = 1

ellipse
center at

\left( \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right)

major axis is

x ^ { \prime }

-axis
vertices at

\left( 4 + \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right)

and

\left( - 4 + \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right)

\theta = 63.4 ^ { \circ }

x ^ { \prime 2 } = - 16 y ^ { \prime }

parabola
vertex at

( 0,0 )

focus at

( 0 , - 4 )

\theta = 63.4 ^ { \circ }

\frac { x ^ { \prime 2 } } { 16 } - \frac { y ^ { \prime 2 } } { 4 } = 1

hyperbola
center at

( 0,0 )

transverse axis is the

x ^ { \prime }

-axis
vertices at

( \pm 4,0 )

Find an Equation of the Parabola Described and State the Two

Multiple Choice