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 are both 12.25 inches tall and stand 70 inches apart. Find the vertical distance from the Roadway to the cable at a point on the road 14 inches from the lowest point of the cable.

Graph the curve whose parametric equations are given. - $x=3 \cos t, y=2 \sin t ; 0 \leq t \leq 2 \pi$ A) B) C) D)

Rotate the axes so that the new equation contains no xy-term. Graph the new equation. - $17 x^{2}-12 x y+8 y^{2}-68 x+24 y-12=0$

Discuss the equation and graph it. - $r = \frac { 4 } { 2 - \sin \theta }$

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola. - $25 y ^ { 2 } - 16 x ^ { 2 } = 400$

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.

Find the vertex, focus, and directrix of the parabola. Graph the equation. - $y^{2}+12 y=4 x-16$ A) vertex: $( - 5 , - 6 )$ focus: $( - 4 , - 6 )$ directrix: $x = - 6$ B) vertex: $( - 5 , - 6 )$ focus: $( - 6 , - 6 )$ directrix: $x = - 4$ C) vertex: $( - 5 , - 6 )$ focus: $( - 5 , - 7 )$ directrix: $y = - 5$ D) vertex: $( - 5 , - 6 )$ focus: $( - 5 , - 5 )$ directrix: $y = - 7$

Name the conic. -

Identify the equation without applying a rotation of axes. - $2 x ^ { 2 } + 8 x y + 16 y ^ { 2 } - 3 x - 2 y + 10 = 0$

Match the equation to the graph. - $( y - 2 ) ^ { 2 } = 6 ( x - 2 )$

Find an equation for the ellipse described. -Center at $( 0,0 ) ;$ focus at $( 0,5 )$ ; vertex at $( 0 , - 7 )$

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. - $3 x ^ { 2 } + 5 x y + 3 y ^ { 2 } - 8 x + 8 y = 0$

Write an equation for the hyperbola. -

Find an equation for the parabola described. -Vertex at (6, 1); focus at (6, 3) A) $( y - 1 ) ^ { 2 } = - 12 ( x - 6 )$ B) $( y - 1 ) ^ { 2 } = 12 ( x - 6 )$ C) $( x - 6 ) ^ { 2 } = 8 ( y - 1 )$ D) $( x - 6 ) ^ { 2 } = - 8 ( y - 1 )$

Discuss the equation and graph it. - $r = \frac { 3 } { 2 + 4 \sin \theta }$

Find an equation for the hyperbola described. Graph the equation. -Center at $( 0,0 ) ;$ focus at $( \sqrt { 13 } , 0 )$ ;vertex at $( 2,0 )$

Match the graph to its equation. -

Identify the equation without completing the square. - $2 y ^ { 2 } - 2 x - 3 y = 0$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equati - $17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0$

Identify the equation without completing the square. - $2 x ^ { 2 } - 4 x + y + 2 = 0$