Question

Eliminate the parameter theta and obtain the standard form of the rectangular equation. hyperbola: x = h a sec(theta), y = k b tan(theta)

Answer 1

To eliminate the parameter θ and obtain the standard form of the rectangular equation for the hyperbola x = ha sec(θ), y = kb tan(θ), you need to express the given parametric equations in terms of x and y. Rearrange the first equation for θ and substitute it in the second equation. Use the trigonometric identity tan(θ) = sin(θ)/cos(θ) to simplify the equation, and rearrange it to obtain the standard form of the rectangular equation.

Step-by-step explanation:

To eliminate the parameter θ and obtain the standard form of the rectangular equation for the hyperbola, we need to express the given parametric equations in terms of x and y.

Given:

x = ha sec(θ)

y = kb tan(θ)

To eliminate the parameter θ, we can rearrange the first equation for θ:

θ = sec-1(x/(ha))

Substituting this value of θ in the second equation:

y = kb tan(sec-1(x/(ha)))

Using the trigonometric identity tan(θ) = sin(θ)/cos(θ), we can further simplify the equation:

y = kb(x/(ha))

Finally, we can rearrange this equation to obtain the standard form of the rectangular equation for the hyperbola:

y = (kb/h)(x/a)