Hi, need some help with tangents and asymptotes

Hi all, I've recently bought a few maths books to study in my spare time and I'm stuck on the below question. I've found the point P and I think the equation of the tangent is y = (4rt3)(x) - 2rt3. I have no idea how to find the equations of the asymptotes.

This question is about a Hyperbola in R^2.

It is described by the below parametric equations.

x = sec t, y = 2 tan t (t ∈ (-π/2 , π/2) U ( π/2 , (3π)/2)

The coordinates of point P, where t = π/3, are ( 2 , 2*sqrt3)

The gradient of the tangent to the hyperbola at the point with parameter t is 2/sin t, where t is not equal to 0 or π. Using this information, ﬁnd the equation of the tangent to the hyperbola at the

point P.

Find the equations of the asymptotes of the hyperbola

So could someone please confirm my answer, for the equation of the tangent and how to get the equations of the asymptotes?

Thanks (Talking)

Re: Hi, need some help with tangents and asymptotes

I think you dropped a 3 in your calculation (or in your typing of your answer). The equation of the tangent line is:

$\displaystyle 3y=4\sqrt{3}x-2\sqrt{3}$

Now as to the asymptotes: Remember, $\displaystyle 1+tan^2(t)=sec^2(t)$, so your hyperbola in standard form is $\displaystyle {x^2\over 1}-{y^2\over 4}=1$ and so the asymptotes are $\displaystyle y=\pm 2x$.

Alternatively, the asymptotes for such a hyperbola are of the form $\displaystyle y=\pm mx$ where m=$\displaystyle \lim_{x\rightarrow \infty}{y\over x}$. Parametrically, $\displaystyle {y\over x}=2sin(t)$ and for x to go to infinity, t must approach $\displaystyle \pi/2$. So $\displaystyle m=\lim_{t\rightarrow \pi/2}2sin(t)=2$.

Re: Hi, need some help with tangents and asymptotes

Thanks for the help Johng, it always seems so much clearer when you have the answer :D