Exam 9: Analytic Geometry

Graph the equation. - $x^{2}=12 y$ A) B) C) D)

Identify the equation without applying a rotation of axes. - $x ^ { 2 } + 8 x y + 16 y ^ { 2 } - 2 x + 4 y - 3 = 0$

Identify the equation without completing the square. - $3 x ^ { 2 } + 4 y ^ { 2 } + 4 x + 2 y = 0$

Find an equation of the parabola described. -Focus at (5, 0); vertex at (0, 0) A) $y = 20 x ^ { 2 }$ B) $x = 20 y ^ { 2 }$ C) $x ^ { 2 } = 20 y$ D) $y ^ { 2 } = 20 x$

Graph the curve whose parametric equations are given. - $x=2 t-1, y=t^{2}+5 ;-4 \leq t \leq 4$

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola. - $\frac { ( x - 4 ) ^ { 2 } } { 25 } - \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$

Find the vertex, focus, and directrix of the parabola with the given equation. - $( y + 1 ) ^ { 2 } = 16 ( x + 3 )$

Find a rectangular equation for the plane curve defined by the parametric equations. -Ron throws a ball straight up with an initial speed of 70 feet per second from a height of 3 feet. Find parametric equations that describe the motion of the ball as a function of time. How long is the ball in the air? When is the Ball at its maximum height? What is the maximum height of the ball? A) $x = 0$ and $y = - 16 t ^ { 2 } + 40 t + 6$ $5.284 \mathrm { sec } , 1.25 \mathrm { sec }$ , 25 feet B) $x = 0$ and $y = - 16 t ^ { 2 } + 40 t + 6$ $2.34 \mathrm { sec } , 1.25 \mathrm { sec }$ , $5.99$ feet C) $x = 0$ and $y = - 16 t ^ { 2 } + 40 t + 6$ $4.679 \mathrm { sec } , 1.25 \mathrm { sec }$ , $181.21$ feet D) $x = 0$ and $y = - 16 t ^ { 2 } + 40 t + 6$ $2.642 \mathrm { sec } , 1.25 \mathrm { sec }$ , 31 feet

Rotate the axes so that the new equation contains no xy-term. Graph the new equation. - $31 x^{2}+10 \sqrt{3} x y+21 y^{2}-144=0$

Find an equation for the ellipse described. -Center at $( 0,0 ) ;$ focus at $( 0 , - 2 )$ ; vertex at $( 0,6 )$

Solve the problem. -The orbit of a planet around a sun is an ellipse with the sun at one focus. The aphelion of a planet is its greatest distance from the sun, its perihelion is its shortest distance, and its mean distance is the length of the semimajor Axis of the elliptical orbit. If a planet has a perihelion of 522.2 million miles and a mean distance of 525 million miles, write an equation for the orbit of the planet around the sun. A) $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 516.9 ^ { 2 } } = 1$ B) $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 518.996 ^ { 2 } } = 1$ C) $\frac { x ^ { 2 } } { 519.004 ^ { 2 } } + \frac { y ^ { 2 } } { 519 ^ { 2 } } = 1$ D) $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 2 \cdot 1 ^ { 2 } } = 1$

Find a rectangular equation for the plane curve defined by the parametric equations. -A baseball player hit a baseball with an initial speed of 160 feet per second at an angle of 40° to the horizontal.The ball was hit at a height of 4 feet off the ground. Find parametric equations that describe the motion of the ball as a function of time. How long is the ball in the air? When is the ball at its maximum height? What is the Distance the ball traveled? A) $x = 114.9 t$ and $y = - 16 t ^ { 2 } + 96.45 t + 3$ $6.059 \mathrm { sec } , 3.014 \mathrm { sec }$ , $1171.774$ feet B) $x = 114.9 t$ and $y = - 16 t ^ { 2 } + 96.45 t + 3$ $5.997 \mathrm { sec } , 3.014 \mathrm { sec }$ , $689.055$ feet C) $x = 122.56 t$ and $y = - 16 t ^ { 2 } + 102.88 t + 4$ $6.469 \mathrm { sec } , 3.215 \mathrm { sec }$ , $792.841$ feet D) $x = 122.56 t$ and $y = - 16 t ^ { 2 } + 102.88 t + 4$ $12.937 \mathrm { sec } , 3.215 \mathrm { sec }$ , $1,585.559$ feet

Match the graph to its equation. -

Solve the problem. -An arch in the form of a semiellipse is 52 ft wide at the base and has a height of 20 ft. How wide is the arch at a height of 12 ft above the base?

Find an equation of the parabola described. -Focus at $( 4,0 ) ;$ vertex at $( 0,0 )$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equati - $31 x^{2}+10 \sqrt{3} x y+21 y^{2}-144=0$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 3 \tan t , y = 2 \sec t ; 0 \leq t \leq 2 \pi$

Find the center, foci, and vertices of the ellipse. - $2 x ^ { 2 } + 5 y ^ { 2 } - 12 x + 60 y + 188 = 0$

Graph the equation. - $(y+1)^{2}=8(x-1)$

Match the equation to the graph. - $( x - 2 ) ^ { 2 } = 7 ( y - 1 )$