Convert the following from rectangular to polar Identify the

Convert the following from rectangular to polar. Identify the curve in (c). (-6,6) (7, -7 squareroot 3) x^2 + y^2 - 4x + 14y = 0 to rectangular. Identify the curve in (c).

Solution

a).

point(-6,6)

in polar x=rcos(theta) =-6

y=rsin(theta) = 6

sin(theta)/cos(theta) = 6/-6

tan(theta) =6/-6 =-1

so theta = 135 degree or 3pi/4

r = sqrt(6^2 +6^2) = sqrt(36+36)

r = sqrt(72)

r = 6sqrt(2)

(r,theta) = (6sqrt(2) , 3pi/4)

b).

(7,-7sqrt(3) )

tan(theta) = -7sqrt(3)/7

tan(theta) = -sqrt(3)

theta = 2pi - pi/3 (since \'x\' co ordinate is positive)

theta = 5pi/3

r = sqrt(7^2 +(7sqrt(3)^2 )

r= sqrt(49 +49*3)

r = 14

(r,theta) = (14 , 5pi/3)

c).

x^2 +y^2 -4x +16y =0

x^2 -4x +4 +y^2 +16y +64 = 4+64

(x-2)^2 + (y+8)^2 = 68

this is curve represents a circle with radius = sqrt(68)

center (2,-8)

to convert it polar plug x=rcos(theta) y =rsin(theta)

r^2 cos^2(theta) +r^2 sin^2(theta) -4rcos(theta) +16rsin(theta) =0

r^2 [ cos^2(theta) +sin^2(theta) - 4rcos(theta) +16rsin(theta) =0

r^2 -4rcos(theta) +16rsin(theta) =0