1. Find the vertex, focus and directrix of the parabola given by y 2 = −6x, and sketch its graph.

(c) r2 = sin θ and r = sin 2θ (d) r = 4 sin(2θ) and r = 2

2. Find the vertex, focus and directrix of the parabola  1 2 given by y = x − 2x + 5 . 4

20. Find the area of the region within the inner loop of the graph of r = 1 + 2 cos θ.

3. Find the equation of the parabola with vertex at (3, 2) and focus at (1, 2).

21. Find the area of the region between the loops of the graph of r = 1 + 2 cos θ.

4. Find an equation for the parabola whose axis is parallel to the y-axis and passes through the points (0, 3), (3, 4), (4, 11).

22. Find the area of the region that lies inside both curves r = sin 2θ and r = sin θ.

5. Find the center, foci, vertices, eccentricity, and directrices of the following.

23. Find the area of the region common to the interiors of the graphs of r = 4 sin θ and r = 2.

(a) 9x + 4y + 36x − 24y + 36 = 0

24. Find the length of the graph r = 1/θ over the interval π ≤ θ ≤ 2π.

(b) 12x2 + 20y 2 − 12x + 40y − 37 = 0

25. Find the length of the following polar curves.

2

2

6. Find an equation for the ellipse with vertices (5,0) and 3 (-5,0) and eccentricity . 5 7. Find an equation of the ellipse with vertices (3,1) and (3,9), and minor axis of length 6. 8. Find the area of the region bounded by the graph of y2 x2 4 + 1 = 1. 9. Find the center, vertices, foci, and directrices of the following. (a)

(x−1)2 4 2

y+2 1

=1

2

(b) 9x − y − 36x − 6y + 18 = 0. 10. Find an equation for the hyperbola with vertices (-1,0) and (1,0) and whose asymptotes are given by y = ±3x. 11. Find an equation of the hyperbola such that for any point on the hyperbola, the difference of its distances from the points (2,2) and (10,2) is 6.

(a) r = 5 cos θ, θ ∈ [0, 3π/4] (b) r = e2θ , θ ∈ [0, 2π] (c) r = θ2 , θ ∈ [0, 2π] 26. For the following, (i) find the eccentricity, (ii) identify the conic, (iii) write an equation of the directrix corresponding to the focus at the pole, and (iv) sketch the curve. (a) r = (b) r = (c) r = (d) r = (e) r =

4 1+cos θ 5 2+sin θ 4 1−3 cos θ 6 3−2 cos θ 1 2+sin θ

(f) r =

9 5−6 sin θ

(g) r =

1 1−2 sin θ

(h) r =

1 5−3 sin θ

(i) r =

10 4+5 cos θ

27. For the following, find a polar equation of the conic having a focus at the pole and satisfying the given conditions

12. For the parametric equations x = 2t, and y = 3t − 1, find dy/dx and d2 y/dx2 and evaluate the two derivatives when t = 3. Do the same for x = 2+sec θ, y = 1+2 tan θ for θ = π/6.

(b) ellipse; e = 1/2; corresponding vertex at (4, π)

13. Find an equation of the tangent line to the graph of x = 2 cot θ, y = 2 sin2 θ at θ = π/4.

(d) hyperbola; vertices at (1, π/2) and (3, π/2)

14. Find all points (if any) of horizontal and vertical tangency on the graph of x = 1 − t, y = t3 − 3t.

(f) parabola; vertex at (6, π/2)

15. Find an equivalent polar equation for the following rectangular equation x2 − 4ay − 4a2 = 0. 16. Find a rectangular equation having the polar equation r = 2−36sin θ . 17. Find the slope of the graph of r = 3(1 − cos θ) at θ = π/2. Do the same for r = θ at θ = π. 18. Find the horizontal and vertical tangent lines to the polar curve r = 1 + sin θ. 19. Find the points of intersection of the given curves. (a) r = 4 − 5 sin θ and r = 3 sin θ (b) r = cos θ and r = 1 − cos θ

(a) parabola; vertex at (4, 3π/2) (c) hyperbola; e = 4/3, r cos θ = 9 is the directrix corresponding to the focus at the pole (e) ellipse; vertices at (3,0) and (1, π)

28. (a) Find a polar equation of the hyperbola having a focus at the pole and the corresponding directrix to the 4 left of the focus if the point (2, π) is on the hyper3 bola and e = 3. (b) Write an equation of the directrix corresponding to the focus at the pole. 29. (a) Find a polar equation of the hyperbola for which e = 3 and which has the line r sin θ = 3 as the directrix corresponding to a focus at the pole. (b) Find the polar equations of the two lines through the pole that are parallel to the asymptotes of the hyperbola. 30. Find the area of the region inside the ellipse r = 3 and above the parabola r = 1+sin θ.

6 2−sin θ

EA Arances

Math 54 exercises 2nd le