Answer
$\dfrac{3 \sqrt 3}{5}$
Work Step by Step
We have: $\dfrac{dr}{ d \theta}=a \sec \theta \tan (2 \theta )$
$\dfrac{dy}{dx}=\dfrac{dy/d \theta }{dx/d \theta }\\=\dfrac{r \cos \theta+\sin\theta \dfrac{dr}{ d \theta}}{- r \sin \theta+\cos\theta \dfrac{dr}{ d \theta}}$
Now,
$\dfrac{dy}{dx}=\dfrac{2a (\dfrac{\sqrt 3}{2}) +\dfrac{1}{2} \times 4 \sqrt 3 a}{-2a \dfrac{1}{2}+\dfrac{\sqrt 3}{2} \times 4 \sqrt 3 a}\\=\dfrac{3 \sqrt 3}{5}$
