Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 3 , y = \frac { t } { t + 3 }$
A) $y = \frac { x - 4 } { x }$
B) $y = \frac { x - 3 } { x }$
C) $y = \frac { x - 2 } { 3 x }$
D) $y = \frac { 2 x - 1 } { 2 x }$
E) none of these

Question

Quizplus · Accepted Answer

We start by solving the equation y = t/(t+3) for t:

t = 3y / (1-y)

Then we substitute into the equation x = t+3:

x = 3y/(1-y) + 3 = (3y + 3 - 3y)/(1-y) = 3/(1-y)

We can then solve for y:

y(1-x) = x-3

y = (x-3)/(1-x)

Multiplying the numerator and denominator of the right-hand side by -1, we obtain:

y = (3-x)/(x-1) = (x-3)/(-x+1) = (x-3)/(1-x)

This matches the form of choice (B), so the answer is B.

Find a Rectangular-Coordinate Equation for the Curve by Eliminating the Parameter