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Math 142 PRINT YOUR NAME: Fall 1999 SIGN YOUR NAME:
Analytic Geometry Final Exam SECTION #:
For problems 1-11, show all your work, and write your answer in the blank provided. Each problem is worth 6 points. You can earn 0, 3, or 6 points on each problem. Sufficient work must be shown to receive credit.
- Convert the polar coordinates (6, 23 π ) to rectangular coordinates.
- Find the directrix of the parabola y^2 = − 6 x.
- Suppose z 1 = 4e−^
πi 6 and z 2 = 2e
2 πi 3 . Compute
z 1 z 2
and express your answer in polar form.
Find the foci of the conic section x^2 −
y^2 9
- Convert the parametric equations x = 2t, y = t^2 − 1, to an equation in x and y only. 5.
- Find the length of the minor axis of the ellipse with center (0, 0), focus (0, 3), and vertex (0, 5). 6.
- Find an appropriate first quadrant angle θ (in radians) so that a rotation of axes by θ transforms the equation 4x^2 + 2
3 xy + 2y^2 = 25 into a new equation of the form au^2 + cv^2 + du + ev + f = 0. (You just need to find θ, not the new equation.)
Find the vertex of the parabola x^2 − 4 x = 2y.
(a) Express the complex number 1 + i in polar form. 9.(a)
(b) Compute (1 + i)^20 and express your answer in standard form a + bi. Be sure to show your work. (b)
- (a)
x^2 4
y^2 9
= 1 (b)
(y + 1)^2 4
− (x − 3)^2 = 1 (c)
x = −3 + sec θ, y = 1 + 2 tan θ
(d)
y^2 9
x^2 4
For problems 14 and 15 below, you must show all of your work in the space provided. Partial credit is possible on these problems. Each problem is worth 9 points.
- Find the equation of the form
(x − h)^2 p^2
(y − k)^2 q^2
= 1 for the ellipse with center (1, −2), vertex (1, 1), and length of minor axis equal to 4.
- Find all four fourth roots of the complex number 16e 4 πi 3 . Write your answers in standard form a + bi, and graph the roots on the complex plane.
