Exam 9: Analytic Geometry

Solve the problem. -An experimental model for a suspension bridge is built in the shape of a parabolic arch. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second Tower. The towers stand 50 inches apart. At a point between the towers and 15 inches along the road from the Base of one tower, the cable is 1 inches above the roadway. Find the height of the towers.

Graph the hyperbola. - $\frac{x^{2}}{4}-\frac{y^{2}}{36}=1$

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. - $11 x ^ { 2 } - 4 x y + 7 y ^ { 2 } - 8 x + 8 y = 0$

Find parametric equations for the rectangular equation. - $y = 3 x ^ { 2 } + 6$

Match the equation to its graph. - $x ^ { 2 } = - 12 y$

Find a polar equation for the conic. A focus is at the pole. - $\mathrm { e } = 5$ ; directrix is perpendicular to the polar axis 3 to the right of the pole

Graph the curve whose parametric equations are given. - $x=2 \sin t, y=2 \cos t ; 0 \leq t \leq 2 \pi$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 5 \cos t , y = - 2 \sin t ; 0 \leq t \leq 2 \pi$

Find the center, foci, and vertices of the ellipse. - $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 64 } = 1$

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. - $x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0$

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

Find an equation for the hyperbola described. -center at $( 9,4 )$ ; focus at $( 7,4 )$ ; vertex at $( 8,4 )$

Write an equation for the graph. -

Discuss the equation and graph it. - $r=\frac{6}{3-3 \cos \theta}$

Find the vertex, focus, and directrix of the parabola with the given equation. - $( x + 1 ) ^ { 2 } = 12 ( y - 4 )$

Match the equation to its graph. - $y ^ { 2 } = 13 x$

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

Solve the problem. -Find parametric equations for an object that moves along the ellipse 1 with the motion described. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 81 } = 1$ The motion begins at $( 3,0 )$ , is counterclockwise, and requires 5 seconds for a complete revolution.

Solve the problem. -The roof of a building is in the shape of the hyperbola $y ^ { 2 } - x ^ { 2 } = 49$ , where $x$ and $y$ are in meters. Determine the distance, $w$ , the outside walls are apart, if the height of each wall is $12 \mathrm {~m}$ .

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