ASSIGNMENT I - MATH 104
KATHMANDU UNIVERSITY
DEPARTMENT OF MATHEMATICS
1. Define polar, cylindrical and spherical coordinates of a point in space. Establish the
relation among Cartesian, cylindrical and spherical coordinates in three dimensional
coordinates.
2. Replace the following Cartesian equations by equivalent polar equations.
x2 y 2
(a) y = −5 (b)
+
= 1 (c) (x − 1)2 + (y − 2)2 = 9 (d) xy = 4
4
4
3. (a) Find the rectangular coordinates for (2, π/2, π/3) and (2, 2π/3, 1).
√
(b) Find the spherical coordinates for (0, 2 3, −2) and the cylindrical coordinates for
(3, −3, −7).
(c) Find the equation in spherical coordinate system.
(i) (x + 1)2 + (y − 1)2 + z 2 = 1.
p
(ii) z = x2 + y 2 .
(d) Find the equation in cylindrical polar coordinates for x2 + (y − 3)2 = 9.
(e) Find the Cartesian and cylindrical coordinate equations for the equation
φ = 5π/6, 0 ≤ ρ ≤ 2 with proper ranges for z in Cartesian and r in cylindrical
coordinates.
4. Find the pair of polar coordinates that label the same point.
(a) (3, 0)
(b) (−3, 0)
(c) (2, 2π/3)
(d) (−3, π)
(e) (−3, 2π)
(f) (−2, −π/3).
5. How is the idea of symmetry of a polar curve r = f (θ) helpful to sketch the graph of
the curve? Write the conditions when the curves r = f (θ) is symmetric about x-axis,
and y-axis. Check the symmetry, sketch the graph of the curve r = sin 2θ, and find
the area covered by the curve.
6. Check the symmetry and sketch the graph of the following equations:
(a) r = −2 sin θ
(b) r = 2 cos θ
(c) r = 1 − cos θ (d) r = −1 + sin θ
(e) r = 5 − 5 sin θ
(f) r = 12 + cos θ
(g) r = 2 + cos θ
(i) r = 2 cos 2θ
(j) r = sin 4θ
(k) r2 = 4 sin 2θ.
7. Find the areas of the following regions:
(a) Inside the lemniscate r2 = 4 sin 2θ.
1
(h) r = 2 − cos θ
(b) Inside one leaf of the four leaved rose r = cos 2θ.
(c) Inside the limacon r = 4 + 2 cos θ.
(d) Inside the cardiod r = 2(1 + cos θ).
(e) Inside the circle r = 1 and outside the cardiod r = 1 − cos θ.
(f) The inner loop of the limacon r = 12 + 24 cos θ.
(g) Shared by the circle r = 2 and cardiod r = 2 − 2 cos θ.
(h) Inside the circle r = 4 cos θ and to the right of the vertical line r = sec θ.
(i) Shared by the circles r = 2a cos θ and r = 2a sin θ.
8. (a) Find the length of the cardioid
(i) r = 1 − cos θ
(ii) The spiral r = θ2 , 0 ≤ θ ≤
√
5.
(b) Find the area of the surface generated by revolving the right hand loop of the
lemniscates r2 = cos 2θ about y-axis.
(c) Find the volume of the solid generated by revolving the lemniscate r2 = a2 cos 2θ
about y-axis.
9. (a) Derive the standard equation of conic sections in polar coordinates? Find the
equation of ellipse with eccentricity e and semimajor axis a.
(b) Find the equation of the circle in polar form whose center is at (r0 , θ0 ). Also, find
the equation for the circles through the origin and which lies on
(i) positive x − axis
(ii) negative y − axis.
(c) Find the the centers of the circles in polar coordinate system and then find their
radii:
(i) r = 5
(ii) r = −2 cos θ
(iii) r = 6 sin θ
(d) Find the equation for the hyperbola with eccentricity 3/2 and directrix x = 2.
25
(e) Find eccentricity, directrix, vertices and centre of the ellipse r =
.
16 + 8 sin θ
10. (a) Find the polar equation of the line passing through P0 (2, π/3).
(b) Find the Cartesian equation of the polar line r cos(θ − 2π/3) = 1.
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