Scribd
Upload
40 views11 pages

Advanced Calculus (MATH 104) - Polar Coordinates: Harish Bhandari

This document discusses polar coordinates and spherical coordinates. It provides examples of converting between rectangular and spherical coordinates, as well as examples of conic sections like parabolas, ellipses, and hyperbolas expressed in polar coordinate equations. It also discusses using polar coordinates to represent lines and circles. Several exercises are provided at the end to sketch conic sections and find polar coordinate equations.

Uploaded by

Bibhushan Gautam
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
40 views11 pages

Advanced Calculus (MATH 104) - Polar Coordinates: Harish Bhandari

This document discusses polar coordinates and spherical coordinates. It provides examples of converting between rectangular and spherical coordinates, as well as examples of conic sections like parabolas, ellipses, and hyperbolas expressed in polar coordinate equations. It also discusses using polar coordinates to represent lines and circles. Several exercises are provided at the end to sketch conic sections and find polar coordinate equations.

Uploaded by

Bibhushan Gautam
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 11

Advanced Calculus(MATH 104)-Polar Coordinates

Harish Bhandari

Department of Mathematics
Kathmandu University
Dhulikhel, Nepal

Lecture 5
August 17, 2019
Relations of spherical coordinates

Harish Bhandari Coordinates systems


Relations of spherical coordinates
Spherical coordinates and rectangular coordinates

Spherical coordinates and Cylindrical coordinates

Harish Bhandari Coordinates systems


Example

1. Find the spherical coordinate equation for


a. x 2 +p
y 2 + (z − 1)2 = 1
b. z = x 2 + y 2
√
2. Find the spherical coordinates for (0, 2 3, −2)

Harish Bhandari Coordinates systems


Conics in polar coordinates

PF = r
FB = rcosθ
PD = k − FB
= k − rcosθ
PF
Also, = e (Constant)
PD
PF = ePD
r = e(k − rcosθ)

e = 1 : Parabola
e < 1 : Ellipse
e > 1 : Hyperbola

Harish Bhandari Coordinates systems


Conics in polar coordinates

Harish Bhandari Coordinates systems


Conics in polar coordinates
Example
1. Find the equation of the conic with one focus at origin:
a. e = 1, x = 2
b. e = 51 , y = −10
25
2. Find the equation of directrix of the parabola: r = 10+10cosθ

Ellipse with eccentricity e and semi-major axis a

a
k= − ae
e
ke = a(1 − e 2 )

∴ Polar equation of ellipse with


eccentricity e and semi-major axis
a is
a(1 − e 2 )
r=
1 + ecosθ
Harish Bhandari Coordinates systems
Example
400
Sketch r = 16+8sinθ

Harish Bhandari Coordinates systems


Equations of Line

Example

√
If θ0 = π4 , r0 = 2, we
obtain that

From right angled triangle π √


OP0 P, rcos(θ − ) = 2
π   π4 √
OP0 = OPcos(∠POP0 ) rcosθcos + rsinθsin = 2
4 4
r0 = rcos(θ − θ0 ) 1 1 √
√ rcosθ + √ rsinθ = 2
where P0 (r0 , θ0 ) is the foot 2 2
of perpendicular of the line x +y =2
L and r0 ≥ 0
Harish Bhandari Coordinates systems
Equations of Circle

b. If the circle’s center


lies on +ve x-axis
From Cosine law,
r = 2acosθ (θ0 = 0)
a2 = r02 +r 2 −2r0 rcos(θ−θ0 ) c. If the circle’s center
lies on +ve y-axis 
, r = 2asinθ θ0 = π2
a. If the circle passes
through the origin,
then r0 = a, so

r = 2acos(θ − θ0 )

Harish Bhandari Coordinates systems


Some Exercises

Lines
π
 √
2π

a. rcos θ − 4 = 2 b. rcos θ − 3 =3
3π π
 
c. rcos θ + 1 4 = d. rcos θ + =2 3
√ √ √
e. 2x + 2y = 6 f. 3x − y = 1
g. y = −5 h. x = −4
Sketch the parabolas and ellipses:
Find a polar equation for
each conic section:

Harish Bhandari Coordinates systems

You might also like