Download MTH U242 QUIZ 9 Answers 1. Convert the following polar

MTH U242
QUIZ 9
Answers
1. Convert the following polar coordinates to rectangular coordinates. (Make sure you’re set to
radians!!)
(a) r = 5, θ = 5π/3
√
x = r cos θ = 5 cos(5π/3) = 5(1/2) = 2.5; y = 5 sin(5π/3) = −5 3/2 ≈ −4.3301
(b) r = 3.2566, θ = 3.2298
x = (3.2566)(cos(3.2298) ≈ −3.2439; y = (3.2566)(sin(3.2298) ≈ −0.2869.
2. Convert the rectangular coordinates to polar. Use positive r and θ.
(a) x = 2, y = 7
p
√
r = x2 + y 2 = 53 ≈ 7.2801. This lies in the first quadrant, so θ = arctan(7/2) ≈ 1.2925.
(b) x = −3, y = −4
p
r = (−3)2 + (−4)2 = 5. This lies in the 3nd quadrant, so θ = π + arctan(4/3) ≈ 4.0689.
3. Consider the polar equation r = 6 cos θ − 4 sin θ.
(a) Convert this into an equation in x and y (HINT: Multiply through by r first.)
2
2
r2 = 6 · |r cos
|{z}
{z θ} − 4 · r| sin
{z θ}, so x + y = 6x − 4y.
x2 +y2
x
y
(b) Show that this is a circle and find its center and radius.
= 0
x2 − 6x + +y 2 + 4y+
2
2
x − 6x + 9 + y + 4y + 4 = 13
³√ ´2
2
2
13
(x − 3) + (y + 2) =
Thus, this is a circle of radius
√
13 with center at (3, −2).
4. Let P = (1, 5, 9) and Q = (2, 2, 3).
(a) Find the distance from P to the xz-plane.
Distance from xz-plane = |y − coordinate| = 5
(b) Find the distance from P to the z-axis.
√
√
Distance from z-axis = distance from (0, 0, 9) = 12 + 52 + 02 = 26 ≈ 5.0990
(c) Find the distance from P to Q.
p
√
By distance formula this is (2 − 1)2 + (2 − 5)2 + (3 − 9)2 = 46 ≈ 6.7823.