If the focus of a parabola is $(1, 0)$ and its directrix is $x + y = 5$,what is its vertex?

The length of the chord of the parabola $y^2 = x$ which is bisected at the point $(2, 1)$ is

Let $C$ be the locus of the mirror image of a point on the parabola $y^{2}=4x$ with respect to the line $y=x$. Then the equation of the tangent to $C$ at $P(2,1)$ is:

$A$ pair of tangents is drawn from an external point $P$ to the parabola $y^2 = 4x$. If $\theta_1$ and $\theta_2$ are the angles made by the tangents with the $x$-axis such that $\theta_1 + \theta_2 = \frac{\pi}{4}$,find the locus of $P$.

The equation of the normal at the point $\left( \frac{a}{4}, a \right)$ to the parabola $y^2 = 4ax$ is:

The equation of a tangent to the parabola $y^2 = 8x$ is $y = x + 2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is