The tangents to the curve $y = (x - 2)^2 - 1$ at its points of intersection with the line $x - y = 3$ intersect at the point:

What is the common tangent to the parabola $y^{2} = 8ax$ and the circle $x^{2} + y^{2} = 2a^{2}$?

The Cartesian coordinates of the point on the parabola $y^2 = -16x$,whose parameter is $t = \frac{1}{2}$,are

Suppose that the equation $f(x) = x^{2} + bx + c = 0$ has two distinct real roots $\alpha$ and $\beta$. The angle between the tangent to the curve $y = f(x)$ at the point $\left(\frac{\alpha + \beta}{2}, f\left(\frac{\alpha + \beta}{2}\right)\right)$ and the positive direction of the $x$-axis is (in $^{\circ}$)

Find the minimum distance from the point $(0, c)$ to the parabola $y = x^2$,where $0 \leq c \leq 5$.

If the angle between the tangents drawn through the point $(-2, -1)$ to the parabola $y^2 = 4x$ is $\theta$,then $\tan 2\theta =$