Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.

Question

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. ​  ​

Quizplus · Accepted Answer

Using the parameter t, we can express the curve in polar coordinates as: 
r = 1 + 2sin(t)
θ = π/4 + 2t 
To eliminate the parameter t, we can use the trigonometric identity: 
sin(t) = (1/2)(2sin(t))
Substituting this into the equation for r, we get: 
r = 1 + r*sin(t) 
r = 1 + 2sin(t) 
r = 1 + 2r*sin(t)/(2r) 
r = 1 + 2y/2 
r = 1 + y 
To find the corresponding rectangular equation, we can use the conversion formulas: 
x = r*cos(θ) 
y = r*sin(θ) 
Substituting the expression for r, we get: 
y = 1 + y*sin(θ) 
y - y*sin(θ) = 1 
y(1 - sin(θ)) = 1 
y = 1/(1 - sin(θ)) 
Since θ = π/4 + 2t, we can substitute back to get the final rectangular equation: 
y = 1/(1 - sin(π/4 + 2t)) 
Or, if we simplify using the trigonometric identity: 
y = 1/(cos(2t)) 
x = r*cos(π/4 + 2t) 
x = (1 + 2sin(t))*cos(π/4 + 2t) 
x = (1 + 2sin(π/4 + 2t))*cos(2t) 
x = (1 + √2*cos(2t))*cos(2t) 
Therefore, the corresponding rectangular equation is: 
y = 1/(cos(2t)) 
x = (1 + √2*cos(2t))*cos(2t) 
Which matches choice B.

Eliminate the Parameter and Write the Corresponding Rectangular Equation Whose