If $y = 2x + 3$ is a tangent to the parabola $y^2 = 24x$,what is the distance between the tangent and the parallel normal?

The maximum area of a circle centered at the origin,which is inscribed in the parabola $y = x^2 - 100$,can be expressed as $\frac{a\pi}{b}$,where $a$ and $b$ are coprime numbers. Then the value of $a + b$ is:

Let $ABCD$ be a trapezium whose vertices lie on the parabola $y^2=4x$. Let the sides $AD$ and $BC$ of the trapezium be parallel to the $y$-axis. If the diagonal $AC$ is of length $\frac{25}{4}$ and it passes through the point $(1,0)$,then the area of $ABCD$ is:

The point on the curve $x^{2}=2 y$ which is nearest to the point $(0,5)$ 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:

If $L(p, q), q > 3$ is one end of the latus rectum of the parabola $(y-2)^2 = 3(x-1)$,then the equation of the tangent at $L$ to this parabola is