Consider a circle that’s centered at the origin and passes through the points B = (2, 2) and C = (−2, 2)...

Question

(a) Find a polar equation whose graph is the above-mentioned arc connecting B and C.

(b) Convert the polar equation you found in part (a) to a rectangular/Cartesian equation.

(c) Find a parametric equation modeling a particle moving on the above-mentioned arc starting at C at t = 0, and ending at B at t = 1.

Answer 1

a) tan(theta) = y/x, theta = tan^-1(y/x) rsin(theta) = y, rcos(theta) = x

for a circle with radius 2sqr2, center at the origin (0,0)

the polar equation is r=2sqr2

b) x^2 + y^2 = (2sqr2)^2=4(2)

x^2 + y^2 = 8 is the equation of a circle centered at the origin through the points (-2,2) and (2,2)

its radius = 2sqr2 = about 2.828

c) for parametic equations, x= a function of cos(t) and y = a function of sin(t),

with t varying from 0 to 1 and the angle theta varying from 3p/4 to pi/4

or from 135 to 45 degrees

x=2sqr2cos[3pi/4-pi(t)/2]

y = 2sq2sin[3pi/4 -pi(t)/2]