Exam 9: Analytic Geometry

Rotate the axes so that the new equation contains no xy-term. Graph the new equation. - $5 x^{2}-6 x y+5 y^{2}-8=0$

Find an equation of the parabola described. -Directrix the line y = 3; vertex at (0, 0) A) $y = - \frac { 1 } { 12 } x ^ { 2 }$ B) $y = - 12 x ^ { 2 }$ C) $x = 3 y ^ { 2 }$ D) $x = - \frac { 1 } { 12 } y ^ { 2 }$

Find an equation for the ellipse described. -Center at $( 0,0 )$ ; focus at $( 0,2 )$ ; vertex at $( 0,3 )$

Solve the problem. -A hall 130 feet in length was designed as a whispering gallery. If the ceiling is 25 feet high at the center, how far from the center are the foci located?

Find an equation for the ellipse described. -Vertices at $( - 4,5 )$ and $( 16,5 )$ ; focus at $( 14,5 )$

Graph the hyperbola. - $(y+1)^{2}-9(x-2)^{2}=9$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = t ^ { 3 } + 1 , y = t ^ { 3 } - 10 ; - 2 \leq t \leq 2$

Find a polar equation for the conic. A focus is at the pole. - $\mathrm { e } = \frac { 1 } { 2 } ;$ directrix is perpendicular to the polar axis 2 to the left of the pole

Graph the hyperbola. - $25 y^{2}-4 x^{2}=100$

Rotate the axes so that the new equation contains no xy-term. Graph the new equation. - $x^{2}+2 x y+y^{2}-8 x+8 y=0$

Find the vertex, focus, and directrix of the parabola. Graph the equation. - $(y+2)^{2}=8(x+3)$ A) vertex: $(2,3)$ focus: $(4,3)$ directrix: $x=0$ B) vertex: (-3,-2) focus: (-1,-2) directrix: x=-5 C) vertex: $(3,2)$ focus: $(3,4)$ directrix: $y=0$ D) vertex: (-3,-2) focus: (-3,0) directrix: y=-4

Convert the polar equation to a rectangular equation. - $r = \frac { 4 } { 4 + \cos \theta }$

Name the conic. -

Find an equation for the hyperbola described. -Vertices at $( 0 , \pm 8 ) ;$ asymptotes at $y = \pm \frac { 4 } { 3 } x$

Write an equation for the parabola. - A) $y ^ { 2 } = 8 x$ B) $x ^ { 2 } = - 8 y$ C) $x ^ { 2 } = 8 y$ D) $y ^ { 2 } = - 8 x$

Rotate the axes so that the new equation contains no xy-term. Graph the new equation. - $x^{2}+x y+y^{2}-3 y-6=0$

Find the vertex, focus, and directrix of the parabola with the given equation. - $( x - 2 ) ^ { 2 } = - 20 ( y - 3 )$

Solve the problem. -A sealed-beam headlight is in the shape of a paraboloid of revolution. The bulb, which is placed at the focus, is 3 centimeters from the vertex. If the depth is to be 6 centimeters, what is the diameter of the headlight at its opening?

Find an equation for the hyperbola described. -Vertices at $( \pm 2,0 )$ ; foci at $( \pm 11,0 )$

Graph the ellipse and locate the foci. - $16 x^{2}+9 y^{2}=144$