A. Correct! These are the corresponding rectangular coordinates.

Transcription

1 Precalculus - Problem Drill 20: Polar Coordinates No. 1 of Find the rectangular coordinates given the point (0, π) in polar (A) (0, 0) (B) (2, 0) (C) (0, 2) (D) (2, 2) (E) (0, -2) A. Correct! These are the corresponding rectangular Given point (0, π), we have r = 0 and θ = π. We know x = r cos θ and y = r sin θ. So, x = r cos θ = 0 and y = r sin θ = 0. The point (0, π) in rectangular coordinates is (0, 0). The correct answer is (A).

2 No. 2 of Find the rectangular coordinates given the point (5, 0) in polar (A) (6, 0) (B) (0, 0) (C) (5, 0) (D) (5, 5) (E) (0, 6) C. Correct! These are the corresponding polar coordinates that happen to be the same! Given point (5, 0) in polar coordinates, we have r = 5 and θ = 0. We know x = r cos θ and y = r sin θ. Recall that cos 0 = 1 and sin 0 = 0. So, x = r cos θ = 5 1 = 5 and y = r sin θ = 5 0 = 0. The point (5, 0) in rectangular coordinates is (5, 0). The correct answer is (C).

3 No. 3 of Transform equation r = 2 cos θ into rectangular form. (A) 3x + 4y = 5 (B) x 2 + y = 2x (C) x + y 2 = 2x (D) x 2 + y 2 = 2x (E) x + y = 2x Review your algebra. Review your algebra. D. Correct! This is the given equation in rectangular form. The given equation in polar coordinates is r = 2 cos θ. Substitute r = (x 2 + y 2 ) into the equation. Also recall that x x cos θ = = r x + y. Therefore: x r = 2cosθ + y = x + y = 2x 2x x + y The correct answer is (D).

4 No. 4 of Given the curve r = 3 in polar coordinates, identify the curve by transforming it into rectangular (A) Line (B) Circle (C) Parabola (D) Ellipse (E) Hyperbola B. Correct! Yes, this equation defines a circle. Recall that r = (x 2 + y 2 ). So, (x 2 + y 2 ) = 3 or x 2 + y 2 = 9. The correct answer is (B).

5 No. 5 of Find the polar coordinates of the point with rectangular coordinates (0, -2). (A) (0, 0) (B) (2, 0) (C) (0, π) (D) ( 3π 2, 2 ) (E) ( 3π 0, 2 ) D. Correct! This is the given point in polar We have x = 0 and y = -2. We know that r x y = + and tan θ = y x. So, r = x + y = 0+ 4 = 2 and tan θ = y x 2 = 0 = undefined Since the point is on the negative y-axis we get θ 3π coordinate point (0, -2) in polar coordinates is ( 2, 2 ). = 3π 2. Therefore, the rectangular The correct answer is (D).

6 No. 6 of Rewrite the equation 2r sin θ = 6 in rectangular form. (A) y = 2 (B) x = 2 (C) y = 3 (D) x = 3 (E) x = 6 C. Correct! This is the given polar equation in rectangular form. We know that x = r cos θ and y = r sin θ. So, 2r sin θ = 2y 2y = 6 y = 3 The correct answer is (C).

7 No. 7 of Eliminate t in the parametric equations x = t and y = 2t and identify the type of curve it defines. (A) Line (B) Parabola (C) Circle (D) Ellipse (E) Hyperbola B. Correct! These parametric equations describe a parabola. Given parametric equations x = tand y = 2t 2 + 4, substitute x for t in the second equation to get y = 2x This is the equation of a parabola with vertex at (0, 4) open in the positive y direction. The correct answer is (B).

8 No. 8 of Express the polar coordinate point ( π 8 2, 4 ) in rectangular (A) (0, 8) (B) (8, 0) (C) (8, 8) (D) (-8, -8) (E) (0, 0) C. Correct! This is the given point in rectangular form. Recall that r x y π = 4 y = + and tan θ =. Given point x ( ) π 4 8 2, 4 π, we have r = 8 2 and 2 θ. Since, tan θ = = 1, we have y = x. Therefore r = x + y = 2x = x 2 which implies that x = 8. This gives y = x = 8. The point ( π 8 2, 4 ) coordinates is (8, 8). The correct answer is (C). in rectangular

9 No. 9 of Rewrite the equation x 2 + y 2 = 4 in polar form. (A) r = 4 (B) r = 0 (C) r = 5 (D) r = 2 (E) r = 1 Review the conversion equations to rewrite the given equation in polar form. Review the conversion equations to rewrite the given equation in polar form. Review the conversion equations to rewrite the given equation in polar form. D. Correct! This is the equation of a circle in polar form. Review the conversion equations to rewrite the given equation in polar form. Recall that x = r cos θ and y = r sin θ. Also, r 2 = x 2 + y 2. It follows that r 2 = 4, which gives r = 2. The correct answer is (D).

10 No. 10 of Eliminate t in the parametric equations x = t 3 and y = 3t 7 and identify the type of curve it defines. (A) Line (B) Parabola (C) Circle (D) Ellipse (E) Hyperbola A. Correct! These parametric equations define a line. x = t 3 t = x + 3 y = 3t 7 t = (y + 7)/3 x + 3 = (y + 7)/3 3x + 9 = y + 7 3x + 2 = y This is the equation of a line. The correct answer is (A).