The length of the latus rectum of the parabola $9x^2 + 16y^2 + 24xy - 4x + 3y = 0$ is

If the tangent to the parabola $y^2 = ax$ makes an angle of $45^{\circ}$ with the $x$-axis,then the point of contact is

If the tangent drawn at the point $P(4,8)$ to the parabola $y^2=16x$ meets the parabola $y^2=16x+80$ at $A$ and $B$,then the mid-point of $AB$ is

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 $^{\circ}$)

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.

If the line $y=2x+k$ is a normal to the parabola $y^2=4x$,then $k=$