Exam 11: Conic Sections

Correct Answer:

Verified

$r = \frac { 0.3 \cdot 7 } { 1 + 0.3 \sin \theta }$ $\Leftrightarrow$ $r = \frac { 21 } { 10 - 3 \sin \theta }$

Correct Answer:

Verified

$3 x y = - 12$ $\Leftrightarrow$ $x y + 4 = 0$ $\Rightarrow$ $B ^ { 2 } - 4 A C = 1$ , so the graph is a hyperbola. To eliminate the $x y$ - term, rotate through $\phi$ , such that $\cot 2 \phi = \frac { A - C } { B } = 0$ $\Leftrightarrow$ $\cos 2 \phi = 0$ , so $\phi = \frac { \pi } { 4 }$ . $x = X \cos \phi - Y \sin \phi = \frac { 1 } { \sqrt { 2 } } X - \frac { 1 } { \sqrt { 2 } } Y$ , $y = X \sin \phi + Y \cos \phi = \frac { 1 } { \sqrt { 2 } } X + \frac { 1 } { \sqrt { 2 } } Y$ . So $Y ^ { 2 } - X ^ { 2 } = 8$

Correct Answer:

Verified

$c = 3$ , since the fociare $( \pm 3,0 )$ , and $a = 4$ since the vertices are $( \pm 4,0 )$ , so $b ^ { 2 } = a ^ { 2 } - c ^ { 2 } = 7$
The major axis is horizontal, so the equation of the ellipse is $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \Leftrightarrow \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$