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Pre-Calculus Pre-AP --Rotation of Axes/Parametric Equations
1) Write parametric equations for the conic
( y − 3)2 − (x + 8)2
7
3
=2
2) For the conic 2 x 2 − 3 xy + 3 y 2 − 4 x − 2 y − 5 = 0 , answer the following:
a) Use the discriminant to determine the type of conic.
b) Determine the angle of rotation for the axes.
c) Use your calculator to generate a table of values and graph the conic on graph paper.
3) At t = 0, two particles begin moving in the coordinate plane in such a manner that their positions are
⎧ x1 (t ) = 12 − 3t
⎧ x 2 (t ) = 2t − 4
and ⎨
.
⎩ y1 (t ) = 2 + t
⎩ y 2 (t ) = 1.5t − 5
described by the equations ⎨
a) Use slope-intercept form to write the rectangular equation for the position of each particle.
b) Find the point of intersection of the paths of the two particles.
c) Find the time when each particle reaches the point of intersection.
d) Find the minimum distance between the two particles and the time at which this occurs.
e) Find the speed of each particle at t = 5.
f) At t = 5, how far is each particle from the origin?
g) At t = 5, how far is each particle from its starting position?
4) The xy coordinate axes are rotated through an angle θ = 30° to create the x ′y ′ coordinate system.
a) Use rotational equations to find the equation of the parabola y = x 2 in the x ′y ′ coordinate system.
b) Find the focus and directrix in the xy coordinate system.
c) Use your answer to part b) to find the focus and directrix in the x ′y ′ system. (HINT: You may want
to refer to your solutions to part B of the Challenge Problem on rotation of axes.) For the directrix,
clear all fractions from your line equation.
d) Use your answer to part c) to verify that your equation from part a) is the set of all points equidistant
from the focus and directrix in the x ′y ′ system. (Yes, Emily -- this time you DO have to square that
trinomial.)
t−4
⎧
⎪ x(t ) = t − 1
5) For the parametric equations ⎨
, t ∈ (1,5]
t −1
⎪ y (t ) =
t+2
⎩
a) State the domain and range.
b) Eliminate the parameter.
⎧ x(t ) = 2 tan t
, identify the domain and range and eliminate the parameter
(
)
y
t
=
t
3
cos
⎩
6) For the parametric equations ⎨
(express your answer in “y =” form). You should be able to do this without using a calculator. Then graph
the relation on your calculator and verify or modify your results.