Find an equation in cylindrical coordinates for the surface represented by each

Tolnaio

Tolnaio

Answered question

2021-09-18

Find an equation in cylindrical coordinates for the surface represented by each rectangular equation.
a.x2+y2=4z2
b.y2=x

Answer & Explanation

casincal

casincal

Skilled2021-09-19Added 82 answers

Step 1
a. Given x2+y2=4z2
It can be compared with the equation of elliptic cone with its axis along z-axis.
z2h2=x2a2+y2b2
Converting rectangular coordinates to cylindrical coordinates.
r2=x2+y2
tan0=yx
z=z
Step 2
so here the equation becomes
x2+y2=4z2
r2=4z2
Step 3
b. Given y2=x
In conversion
x=rcos0
y=rsin0
z=z
Step 4
Substitute values of x and y in equation.
(rsin0)2=rcos0
r2sin20=rcos0
r(rsin20cos0)=0 divide both sides by r
rsin20cos0=0
r=cos0sin20
r=cosec θcotθ

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