A curve with polar equation r =43/ (3sinθ+19cosθ) represents a line.

Question

This line has a Cartesian equation of the form y=mx+b, where m and b are constants. Give the formula for y in terms of x. For example, if the line had equation y=2x+3 then the answer would be 2⋅x+3

Answer 1

Problem:

A curve with polar equation r =43/ (3sin(t)+19cos(t)) represents a line.

This line has a Cartesian equation of the form y=mx+b, where m and b are constants.

Give the formula for y in terms of x. For example, if the line had equation y=2x+3 then the answer would be 2·x+3

Solution:

r = 43 /(3sin(t) + 19cos(t))

Formulas:

x = r*cos(t)

y = r*sin(t)

r^2 = x^2 + y^2

cos(t) = x/r

sin(t) = y/r

Substitute into the equation to get:

r = 43 /(3sin(t) + 19cos(t))

r = 43 /(3(y/r) + 19(x/r))

r = [43*r] /[3y + 19x]

Divide both sides by r to get:

1 = 43 /[3y + 19x]

3y + 19x = 43

3y = -19x + 43

y = -(19/3)x + 43/3

THEREFORE

-(19/3)x + 43/3