The equation of a tangent to the parabola $y^2 = 4ax$ making an angle $\theta$ with the $x$-axis is

If the line $y = 2x + k$ is a normal to the parabola $y^2 = 4x$ at the point $(t^2, 2t)$,then:

The angle of intersection between the curves $x^2 = 8y$ and $y^2 = 8x$ at the origin is

If the focus of a parabola is $(3, 0)$ and the length of its latus rectum is $8$,what is its vertex?

Let $y=f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y =-\frac{1}{2}$. Then $S=\left\{x \in R : \tan ^{-1}\left(\sqrt{f(x)}+\sin ^{-1}(\sqrt{f(x)+1})\right)=\frac{\pi}{2}\right\}$:

$A$ beam is supported at its ends by supports which are $12 \, m$ apart. Since the load is concentrated at its centre,there is a deflection of $3 \, cm$ at the centre and the deflected beam is in the shape of a parabola. How far from the centre is the deflection $1 \, cm$?