Exam 10: Conic Sections and Analytic Geometry

Use point plotting to graph the plane curve described by the given parametric equations. - $x=2 \tan t \cdot y=3 \sec t: 0 \leq t \leq 2 \pi$

Additional Concepts - $x = 2 t , y = t ^ { 2 } + t + 2$

Identify Conics Without Rotating Axes - $x ^ { 2 } + 12 x y + 36 y ^ { 2 } - 4 x - 3 y + 1 = 0$

Tech: Conic Sections in Polar Coordinates - $r = \frac { 3 } { 1 - \cos \left( \theta - \frac { \pi } { 4 } \right) }$

Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term. - $4 x ^ { 2 } + 5 x y + 4 y ^ { 2 } - 8 x + 8 y = 0$

Graph Hyperbolas Centered at the Origin - $y=\pm \sqrt{x^{2}-3}$

Eliminate the Parameter - $\mathrm { x } = 2 \mathrm { t } - 1 , \mathrm { y } = \mathrm { t } ^ { 2 } + 3 ; - 4 \leq \mathrm { t } \leq 4$

Conic Sections in Polar Coordinates 1 Define Conics in Terms of a Focus and a Directrix - $\mathrm { r } = \frac { 3 } { 1 - 3 \sin \theta }$

Write Equations of Rotated Conics in Standard Form - $31 x ^ { 2 } + 10 \sqrt { 3 } x y + 21 y ^ { 2 } - 144 = 0$

Identify Conics Without Rotating Axes - $2 x ^ { 2 } + 4 x y + 4 y ^ { 2 } + 2 x - 4 y + 2 = 0$

Additional Concepts - $x = - ( y - 10 ) ^ { 2 } - 6$

Convert the equation to the standard form for a hyperbola by completing the square on x and y. - $x ^ { 2 } - y ^ { 2 } + 6 x - 2 y + 7 = 0$

Additional Concepts - $y = x ^ { 2 } + 12 x + 35$

Convert the equation to the standard form for a parabola by completing the square on x or y as appropriate. - $y ^ { 2 } + 6 y - 5 x + 14 = 0$

Find the standard form of the equation of the ellipse satisfying the given conditions. -Major axis horizontal with length 20 ; length of minor axis $= 12 ;$ center $( 0,0 )$

Find Parametric Equations for Functions -The line segment starting at $( - 6 , - 3 )$ with $t = 0$ and ending at $( - 21 , - 9 )$ with $t = 3$

Additional Concepts - $y ^ { 2 } + 6 y - x + 5 = 0$

Graph the Polar Equations of Conics - $r = \frac { 15 } { 5 + 10 \sin \theta } \quad$ Identify the directrix and vertices.

Eliminate the Parameter -A circle: $x = 2 + 5 \cos t , y = 5 + 5 \sin t$

Graph the Polar Equations of Conics - $r = \frac { 9 } { 3 - \sin \theta } \quad$ Identify the directrix and vertices.