Which of the following equations in parametric form represent a hyperbola?

For the ellipse $\frac{x^2}{{\sec }^2\alpha }+\frac{y^2}{{\tan }^2\alpha }=1, \alpha \in \left(0,\frac{\pi }{2}\right)$, which of the following vary as $\alpha$ varies?

Equation of a common tangent to the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ are

If $12x=k$ is the directrix of parabola $y^2+4y+3x=0,$ then which of the following are true?

The mirror image of the parabola $y^2=4x$ in the tangent at $(1, 2)$ to it, is

The following line(s) is (are) tangent to the curve $y=x^2-x$