Exam 9: Conics, Systems of Nonlinear Equations and Inequalities, and Parametric Equations

Find the maximum and minimum values of the ellipse.

Match the equation to the parabola.

(a) Identify the type of conic by the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY term) by rotating the x- and y-axes by an angle of θ to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).

For the polar equation, (a) Identify the conic as either a parabola, ellipse, or hyperbola; (b) find the eccentricity and vertex (or vertices); and (c) graph.

Find an equation of the form Ax² + Bx + Cy + D = 0 or Ay² + By + Cx + D = 0 for the parabola with focus (9, -1) and directrix y = 1.

Recall that the flight of a projectile can be modeled with the parametric equations: , where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet. A baseball is hit at an initial speed of 115 mph and angle of 32° at a height of 2 feet above the ground. If home plate is 425 feet from the back fence, which is 12 feet tall, will the baseball clear the back fence for a home run?

Solve the system of equations by applying the substitution method. X² - y² = 17 Xy = 72

A bridge with a parabolic shape has an opening 122 feet wide at the base of the bridge (where the bridge meets the water), and the height in the center of the bridge is 34 feet. A sailboat whose mast reaches 25 feet above the water is traveling under the bridge 15 feet from the center of the bridge. Will it clear the bridge without scraping its mast? Justify your answer.

Find the general equation of the hyperbola from the given graph.

If the xy-coordinate axes are rotated 45°, find the XY coordinates of the point (x, y) = (-33, 3).

Identify the conic section given by the following equation as a parabola, ellipse, circle, or hyperbola. 64x² + 36y² = 81

For the polar equation, (a) Identify the conic as either a parabola, ellipse, or hyperbola; (b) find the eccentricity and vertex (or vertices); and (c) graph.

Find the general form of the equation of the ellipse with the given graph.

Graph the ellipse.

Recall that the flight of a projectile can be modeled with the parametric equations: , where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet. A projectile is launched at a speed of 520 ft/sec at an angle of 31° with the horizontal. How far does the projectile travel (what is the horizontal distance), and what is the maximum altitude? Round to the nearest integer.

Find the standard form of the ellipse with the following equation. 25x² + 16y² - 50x + 128y = 119

Graph the curve defined by the parametric equations.

(a) Identify the type of conic by the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY term) by rotating the x- and y-axes by an angle of θ to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).

Find the equation of the hyperbola from the given graph.

Graph the ellipse.