Find the Cartesian form of the parametric equation.
x = (2a)(cot T)
y = (2a)(sin^2 T)
how? lol
here's what i got
y = (sin^2 T)
y = (2a)y^2
y = 2a
then?
do the same for X?
i'm stuck there
Proceed to eliminate T from the two equations, you will end up with a single equation involving x and y. Solve for y.
x=(2a)cot(T)....(1a)
x² = (4a²)cot²(T)...(1b)
Using cot²(x)+1 = csc²(x)
we get cot²(x)=csc²(x)-1
1(b) becomes
x² = (4a²)(csc²(T)-1)
or
sin²(T) = 4a²/(4a²+x²).....(1c)
From
y = (2a)(sin^2 T)
we get
sin²(T) = y/(2a) ....(2a)
Substitute (2a) in (1c)
y/(2a) = 4a²/(4a²+x²)
y=8a³/(4a²+x²)
how'd you get (4a²+x²) from
x² = (4a²)(csc²(T)-1) ?
i know you can x² = sin²(T) but
how'd you get
(4a²+x²)??
By moving the "-1" term to the left hand side, we end up with only one term containing T:
x² = (4a²)(csc²(T)-1)
x² = 4a²/sin²(T) - 4a²
x²+4a² = 4a²/sin²(T)
sin²(T) = 4a²/(x²+4a²)
To find the Cartesian form of the given parametric equations x = 2a(cot T) and y = 2a(sin^2 T), you can follow these steps:
Step 1: Solve for cot T in terms of x:
From the equation x = 2a(cot T), divide both sides by 2a to isolate cot T:
cot T = x / (2a)
Step 2: Solve for sin^2 T in terms of y:
From the equation y = 2a(sin^2 T), divide both sides by 2a to isolate sin^2 T:
sin^2 T = y / (2a)
Step 3: Use the identity tan^2 T + 1 = sec^2 T to eliminate cot T from equation 1:
Start with: cot T = x / (2a)
Square both sides: (cot T)^2 = (x^2 / (2a)^2)
Use the identity: cot^2 T = 1 / tan^2 T
Therefore: 1 / tan^2 T = (x^2 / (2a)^2)
Rearrange: tan^2 T = (2a)^2 / x^2
Take the reciprocal of both sides: 1 / tan^2 T = x^2 / (2a)^2
Step 4: Combine equations 2 and 3:
From equation 4, we have: 1 / tan^2 T = x^2 / (2a)^2
From equation 2, we have: sin^2 T = y / (2a)
Combine and rearrange the equations:
1 = x^2 / (2a)^2 + y / (2a)
Step 5: Multiply both sides by (2a)^2 to eliminate fractions:
(2a)^2 = x^2 + (2a)y
4a^2 = x^2 + 2ay
Finally, the Cartesian form of the parametric equations is:
x^2 + 2ay = 4a^2