Write the corresponding rectangular equation for the curve represented by the parametric equations by eliminating the parameter.

Question

Write the corresponding rectangular equation for the curve represented by the parametric equations  by eliminating the parameter. ​

Quizplus · Accepted Answer

To eliminate the parameter t, we solve for t in terms of x and replace it in the equation for y.
From the given parametric equation, we have:
x = 11e^t
y = -4sin(t) + 7cos(t)
Squaring both sides of the second equation and using the identity sin^2(t) + cos^2(t) = 1, we get:
y^2 = 16sin^2(t) + 49cos^2(t) - 56sin(t)cos(t)
Dividing both sides by 16cos^2(t), we get:
(y^2/49) - (56/784)(y/7)(sin(t)/cos(t)) + (sin^2(t)/16cos^2(t)) = 1
Using the trigonometric identity tan(t) = sin(t)/cos(t), we get:
(y^2/49) - (8/49)(y/7)(tan(t)) + (tan^2(t)/1 + tan^2(t)) = 1
Substituting x = 11e^t and solving for t, we get:
t = ln(x/11)
Substituting this in the equation above and simplifying, we get the rectangular equation:
(x^2/121) - (8/11)(xy/77) + (y^2/256) = 1
Which simplifies to:
Answer: C
 
To eliminate the parameter t, we solve for t in terms of x and replace it in the equation for y.
From the given parametric equation, we have:
x = 11e^t
y = -4sin(t) + 7cos(t)
Squaring both sides of the second equation and using the identity sin^2(t) + cos^2(t) = 1, we get:
y^2 = 16sin^2(t) + 49cos^2(t) - 56sin(t)cos(t)
Dividing both sides by 16cos^2(t), we get:
(y^2/49) - (56/784)(y/7)(sin(t)/cos(t)) + (sin^2(t)/16cos^2(t)) = 1
Using the trigonometric identity tan(t) = sin(t)/cos(t), we get:
(y^2/49) - (8/49)(y/7)(tan(t)) + (tan^2(t)/1 + tan^2(t)) = 1
Substituting x = 11e^t and solving for t, we get:
t = ln(x/11)
Substituting this in the equation above and simplifying, we get the rectangular equation:
(x^2/121) - (8/11)(xy/77) + (y^2/256) = 1
Which simplifies to:
7x^2 - 88xy + 121y^2 = 26816
or
Answer: C

Write the Corresponding Rectangular Equation for the Curve Represented by the Parametric