If the line $x + \alpha y + \beta = 0$ touches the curve $4x^3 + 4y^3 = xy(xy + 16)$ at points $(x_1, y_1)$ and $(x_2, y_2)$,where $x_1 \neq x_2$,then the value of $\alpha + \beta$ is (given $\alpha, \beta \in R$).

The equation of the common tangent of the parabolas $x^2=108y$ and $y^2=32x$ is

If $PQ$ is a focal chord of the parabola $y^2=4x$ with focus $S$ and $P=(4,4)$,then $SQ=$

Let chord $PQ$ of length $3\sqrt{13}$ of the parabola $y^2 = 12x$ be such that the ordinates of points $P$ and $Q$ are in the ratio $1:2$. If the chord $PQ$ subtends an angle $\alpha$ at the focus of the parabola, then $\sin \alpha$ is equal to:

The equation of the tangent to the parabola $y^2=8x$,which is parallel to the line $4x-y+3=0$,is

What is the area of the triangle formed by the vertex of the parabola $x^2 = 12y$ and the endpoints of its latus rectum (in square units)?