1. Find the vertex, focus and directrix of the parabolagiven by y2 = −6x, and sketch its graph.

2. Find the vertex, focus and directrix of the parabola

given by y =1

4

(x2 − 2x+ 5

).

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

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

5. Find the center, foci, vertices, eccentricity, and directri-ces of the following.

(a) 9x2 + 4y2 + 36x− 24y + 36 = 0

(b) 12x2 + 20y2 − 12x+ 40y − 37 = 0

6. Find an equation for the ellipse with vertices (5,0) and

(-5,0) and eccentricity3

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 ofx2

4 + y2

1 = 1.

9. Find the center, vertices, foci, and directrices of the fol-lowing.

(a) (x−1)2

4 − y+21 = 1

(b) 9x2 − y2 − 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 anypoint on the hyperbola, the difference of its distancesfrom the points (2,2) and (10,2) is 6.

12. For the parametric equations x = 2t, and y = 3t − 1,find dy/dx and d2y/dx2 and evaluate the two derivativeswhen t = 3. Do the same for x = 2+sec θ, y = 1+2 tan θfor θ = π/6.

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

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

15. Find an equivalent polar equation for the following rect-angular equation x2 − 4ay − 4a2 = 0.

16. Find a rectangular equation having the polar equationr = 6

2−3 sin θ .

17. Find the slope of the graph of r = 3(1 − cos θ) atθ = π/2. Do the same for r = θ at θ = π.

18. Find the horizontal and vertical tangent lines to thepolar 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

(c) r2 = sin θ and r = sin 2θ

(d) r = 4 sin(2θ) and r = 2

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

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

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

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

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

25. Find the length of the following polar curves.

(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) identifythe conic, (iii) write an equation of the directrix corre-sponding to the focus at the pole, and (iv) sketch thecurve.

(a) r = 41+cos θ

(b) r = 52+sin θ

(c) r = 41−3 cos θ

(d) r = 63−2 cos θ

(e) r = 12+sin θ

(f) r = 95−6 sin θ

(g) r = 11−2 sin θ

(h) r = 15−3 sin θ

(i) r = 104+5 cos θ

27. For the following, find a polar equation of the conichaving a focus at the pole and satisfying the given con-ditions

(a) parabola; vertex at (4, 3π/2)

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

(c) hyperbola; e = 4/3, r cos θ = 9 is the directrixcorresponding to the focus at the pole

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

(e) ellipse; vertices at (3,0) and (1, π)

(f) parabola; vertex at (6, π/2)

28. (a) Find a polar equation of the hyperbola having a fo-cus at the pole and the corresponding directrix to the

left of the focus if the point (2,4

3π) is on the hyper-

bola and e = 3. (b) Write an equation of the directrixcorresponding to the focus at the pole.

29. (a) Find a polar equation of the hyperbola for whiche = 3 and which has the line r sin θ = 3 as the directrixcorresponding to a focus at the pole. (b) Find the po-lar equations of the two lines through the pole that areparallel to the asymptotes of the hyperbola.

30. Find the area of the region inside the ellipse r = 62−sin θ

and above the parabola r = 31+sin θ .

EA Arances