What is the equation of the tangent to the parabola $y^2 = 4x$ at the point $(1, 2)$?

If $(at^2, 2at)$ are the coordinates of one end of a focal chord of the parabola $y^2 = 4ax$,find the coordinates of the other end.

The points of intersection of the parabolas $y^2=5x$ and $x^2=5y$ lie on the line

Consider the parabola $y^{2}=4x$. Let $P$ and $Q$ be points on the parabola where $P(4, -4)$ and $Q(9, 6)$. Let $R$ be a point on the arc of the parabola between $P$ and $Q$. Then,the area of $\Delta PQR$ is largest when

Let one root of the quadratic equation in $x$: $(k^2 - 15k + 27)x^2 + 9(k-1)x + 18 = 0$ be twice the other. Then the length of the latus rectum of the parabola $y^2 = 6kx$ is equal to:

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: