Rewrite the parametric equations without the parameter t:
x=sect and y=4tant
A. x^2+y^2/4=1
B. x^2-y^2/16=1
C. x^2/16+y^2=1
D. x^2-y^2=16
note that sec^2 t = 1 + tan^2 t
So,
x^2 = (y/4)^2 + 1
now finish it off
From that, I got x^2=y^2/16+1. Than, to isolate the 1, I subtracted y^2/16 from each side of the equation to get B: x^2-y^2/16=1.
@ Oobleck, thank you for your help.
To rewrite the parametric equations x=sect and y=4tant without the parameter t, we can use trigonometric identities to express x and y in terms of other variables.
Starting with x=sect, we can use the identity sec^2θ = 1 + tan^2θ to rewrite it as:
x = 1/cosθ
Similarly, for y=4tant, we can use the identity tan^2θ = sec^2θ - 1 to rewrite it as:
y = 4(1/cosθ - 1)
Now, let's manipulate these equations to eliminate the parameter θ and express x and y in terms of each other:
From the equation x = 1/cosθ, we can multiply both sides by cosθ to get:
x * cosθ = 1
Next, let's solve the equation y = 4(1/cosθ - 1) for 1/cosθ:
y = 4(1/cosθ - 1)
y = 4/cosθ - 4
y + 4 = 4/cosθ
Now, we can substitute 4/cosθ for y + 4 in the equation x * cosθ = 1:
x * cosθ = y + 4
x * cosθ = 4/cosθ
x * cos^2θ = 4
Using the identity cos^2θ = 1 - sin^2θ, we can rewrite the equation as:
x * (1 - sin^2θ) = 4
x - x * sin^2θ = 4
x - (y + 4) * sin^2θ = 4
Finally, let's simplify this equation:
x - (y + 4) * sin^2θ = 4
x - y * sin^2θ - 4 * sin^2θ = 4
x - y * sin^2θ = 4 + 4 * sin^2θ
The rewritten parametric equations without the parameter t are:
x - y * sin^2θ = 4 + 4 * sin^2θ.
Comparing this equation to the given answer choices:
A. x^2 + y^2/4 = 1
B. x^2 - y^2/16 = 1
C. x^2/16 + y^2 = 1
D. x^2 - y^2 = 16
The correct answer is D. x^2 - y^2 = 16.
To rewrite the parametric equations without the parameter t, we need to eliminate t by expressing it in terms of x and y.
Using the trigonometric identity sec^2(t) = 1 + tan^2(t), we can rewrite the given equations as:
x = sec(t) => 1 + tan^2(t) = x^2 .......(Equation 1)
y = 4tan(t) .......(Equation 2)
Now let's solve Equation 1 for tan^2(t):
tan^2(t) = x^2 - 1
Substituting this into Equation 2:
y = 4tan(t) = 4(sqrt(x^2 - 1))
Squaring both sides of the equation:
y^2 = 16(x^2 - 1) = 16x^2 - 16
Rearranging the equation:
16x^2 - y^2 = 16
Thus, rewriting the parametric equations without the parameter t gives the equation:
D. x^2 - y^2 = 16