91Ó°ÊÓ

Find a viewing window that shows a complete graph of the curve. $$x=t^{2}-4, \quad y=t / 2, \quad-2 \leq t \leq 3$$

Plot the point whose polar coordinates are given. $$(-2,2 \pi / 3)$$

Assume that the graph of the equation is a nondegenerate conic section. Without graphing, determine whether the graph an ellipse, hyperbola, or parabola. $$17 x^{2}-48 x y+31 y^{2}+50=0$$

List four other pairs of polar coordinates for the given point, each with a different combination of signs (that is, \(r > 0, \theta > 0 ; r > 0, \theta < 0 ; r < 0, \theta > 0 ; r < 0, \theta < 0)\). $$(-3,7 \pi / 6)$$

Find the eccentricity of the conic whose equation is given. $$\frac{(x-6)^{2}}{10}-\frac{y^{2}}{40}=1$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$23 x^{2}+26 \sqrt{3} x y-3 y^{2}-16 x+16 \sqrt{3} y+128=0$$

Use Exercise 44 to find a parameterization of the line segment joining the two points. Confirm your answer by graphing. $$(-6,12) \text { and }(12,-10)$$

A comet travels in a parabolic orbit with the sun as focus. When the comet is 60 million miles from the sun, the line segment from the sun to the comet makes an angle of \(\pi / 3\) radians with the axis of the parabolic orbit. Using the sun as the pole and assuming the axis of the orbit lies along the polar axis, find a polar equation for the orbit.

Halley's Comet has an elliptical orbit, with eccentricity .97 and the sun as a focus. The length of the major axis of the orbit is 3364.74 million miles. Using the sun as the pole and assuming the major axis of the orbit is perpedicular to the polar axis, find a polar equation for the orbit.

Two listening stations that are 1 mile apart record an explosion. One microphone receives the sound 2 seconds after the other does. Use the line through the microphones as the \(x\) -axis, with the origin midway between the microphones, and the fact that sound travels at 1100 feet per second to find the equation of a hyperbola on which the explosion is located. Can you determine the exact location of the explosion?