91Ó°ÊÓ

True or False: The distance between two points \((a, b)\) and \((c, d)\) is given by the formula $$ d=\sqrt{(a-c)^{2}+(b-d)^{2}} $$.

Find the center, rertices, foci, and asymptotes of the hyperbola that satisfies the given equation, and sketch the hyperbola. $$4 x^{2}-9 y^{2}=36$$

Identify and graph the conic section given by each of the equations. $$r=\frac{4}{2+5 \cos \theta}$$

Find the vertex and focus of the parabola that satisfies the given equation. Write the equation of the directrix,and sketch the parabola. $$-3 x^{2}=16 y$$

Sara kicks a soccer ball from the ground with an initial velocity of 120 feet per second at an angle of \(30^{\circ}\) to the horizontal. (a) Find the parametric equations that give the position of the ball as a function of time. (b) When is the ball at its maximum height, to the nearest hundredth of a second? What is its maximum height, to the nearest tenth of a foot? (c) How far did the ball travel? Round your answer to the nearest foot.

Find the center, rertices, foci, and asymptotes of the hyperbola that satisfies the given equation, and sketch the hyperbola. $$8 x^{2}-32 x-y^{2}-6 y=41$$

Determine the equation in standard form of the hyperbola that satisfies the given conditions. -Vertices at (0,2),(0,-2)\(;\) foci at (0,3),(0,-3)

Suppose that a satellite receiver with a parabolic cross-section is 36 inches across and 16 inches deep. How far from the vertex must the receptor unit be located to ensure that it is at the focus of the parabola?

The orbit of the moon around Earth is an ellipse, with Earth at one focus. If the major axis of the orbit is 477,736 miles and the minor axis is 477,078 miles, find the maximum and minimum distances from Earth to the moon.

In this set of exercises, you will use hyperbolas to study real-world problems. Astronomy The path of a certain comet is known to be hyperbolic, with the sun at one focus. Assume that a space station is located 13 million miles from the sun and at the center of the hyperbola, and that the comet is 5 million miles from the space station at its point of closest approach. Find the equation of the hyperbola if the coordinate system is set up so that the sun lies on the \(x\) -axis and the origin coincides with the center of the hyperbola.