The parametric curve:
x(t) = 9cos(t)
y(t) = 3sin(2t)
intersects itself at which point? What are the two slopes to the tangent at this point?
Did you graph it?. You have a leminscate. It looks like the 'infinity' symbol.
The slope is given by $\displaystyle \frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}$
$\displaystyle y'(t)=6cos(2t)$
$\displaystyle x'(t)=-9sin(t)$
$\displaystyle \frac{6cos(2t)}{-9sin(t)}$
To find the slope at its intersection point, plug in $\displaystyle t=\frac{\pi}{2}$