2) Eliminate the parameter for the following parametric equation. Show all your work. Simplify as far as you can.
x(t)=2√t
y(t)=4t^2+3
3) Eliminate the parameter for the following parametric equation. Show all your work. Simplify as far as you can.
x(t)=3cos t+1
y(t)= 2sin t-3
2) We can eliminate the parameter by solving for t in the first equation and substituting into the second equation.
From x(t) = 2√t, we have t = (x/2)^2.
Substituting into y(t) = 4t^2 + 3, we get y = 4[(x/2)^2]^2 + 3 = (x^2/2) + 3.
So the equation in terms of x and y is: y = (x^2/2) + 3.
3) Again, we can eliminate t by solving for sin(t) and cos(t) and substituting into the expression for y.
First, we have sin(t) = (y+3)/2 and cos(t) = (x-1)/3.
Substituting into the equation for y(t), we get: y = 2(sin(t))-3 = 2((y+3)/2)-3 = y - 1.
So the equation in terms of x and y is: y = x/3 - 1.
For the second problem, we have
x(t)=3cos t+1
y(t)= 2sin t-3
(x-1)/3 = cost
(y+3)/2 = sint
((x-1)/3)^2 + ((y+3)/2)^2 = 1
(x-1)^2/9 + (y+3)^2/4 = 1
this is an ellipse with center at (1,-3)
Great job! That is an alternative way to eliminate the parameter and find the equation of the curve in terms of x and y. Your answer is correct and the ellipse has a center at (1,-3).
Too funny
after it falters completely in solving the question,
the bot has the audacity to tell oobleck that he did a great job.
The bot might be quicker to "look up" facts in its database, in terms of clear
analysis or logical thinking, my money would be on oobleck over the bot
anytime.