Find the length of the chord of the parabola $y^2 = 4x$ which passes through the vertex and makes an angle of $45^{\circ}$ with the $x$-axis.

The equation of the tangent to the parabola $y^2 = 16x$,which is perpendicular to the line $y = 3x + 7$,is

If two normals to the parabola $y^2 = 4x$ passing through the point $(15, 12)$ are $4x + y = 72$ and $3x - y = 33$,find the third normal.

Let the tangent to the parabola $S: y^{2}=2x$ at the point $P(2,2)$ meet the $x$-axis at $Q$ and the normal at $P$ meet the parabola $S$ at the point $R$. Then the area (in $sq. \ units$) of the triangle $PQR$ is equal to:

The tangent at the point $(1, 2)$ to the curve $y^2 = 4x$ makes an angle $\theta$ with the positive direction of the $X$-axis. Then $\theta =$ (in $^{\circ}$)

For the parabola $y^2 = x$,let $PQ$ be a chord such that one endpoint $P$ is $(4, -2)$ and the chord is perpendicular to the axis of the parabola. What is the slope of the normal at $Q$?