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MTH201 Multivariable Calculus Assignment No 02 Solution & Discussion Due Date: 17-07-2014


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QUESTION NO. 2

SOLUTION:

1) Symmetry about the Initial Line

If the equation of a curve remains unchanged when (r,θ) 
is replaced by either (r,-θ) in its equation ,then the curve 
is symmetric about initial line.

(ii) Symmetry about the y-axis

If when (r, θ) is replaced by either (r,π θ − ) in 
The equation of a curve and an equivalent equation 
is obtained ,then the curve is symmetric about the 
line perpendicular to the initial i.e, the y-axis

(ii) Symmetry about the Pole

If the equation of a curve remains unchanged

when either (-r, θ) or is substituted for (r, θ) 
in its equation ,then the curve is symmetric 
about the pole. In such a case ,the center of 
the curve.

Answer # 1:

Equation of the curve in polar co-ordinates is r2=-4sin2θ

(i) Symmetry about the Initial Line.
r2=-4sin2θ
Remember that sine is an odd function 
Put (r,-θ) in equation 
r2=-4sin2 (-θ)
r2=-4sin (-2θ)
r2=4sin2θ
This equation is not symmetry about initial line 
(ii) Symmetry about the Pole
r2=-4sin2θ
Put (-r, θ) in equation
(-r) 2 =-4sin2θ
r2=-4sin2θ
This equation is symmetry about pole

Q:1:- Equation of the curve in polar co-ordinates is =-4sin2 discuss the symmetry of graph of this curve about.
Initial line
Pole

Equation of the curve in polar co-ordinates is
=-4sin2θ
Symmetry about the Initial Line.
=-4sin2θ
sin is odd function.
Put (r, -θ) in equation 
=-4sin2 (-θ)
=-4sin (-2θ)
=4sin2θ
this equation is not symmetry about initial line 
Symmetry about the Pole
=-4sin2θ
Put (-r, θ) in equation
=-4sin2θ
=-4sin2θ 
This equation is symmetry about pole

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