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

If $y=mx+4$ is a tangent to both the parabolas $y^{2}=4x$ and $x^{2}=2by$,then $b$ is equal to:

The equation of the chord of contact of the tangents drawn from the point $(2, 3)$ to the parabola $y^2 + x = 0$ is:

If the line $x + \alpha y + \beta = 0$ touches the curve $4x^3 + 4y^3 = xy(xy + 16)$ at points $(x_1, y_1)$ and $(x_2, y_2)$,where $x_1 \neq x_2$,then the value of $\alpha + \beta$ is (given $\alpha, \beta \in R$).

The point on the curve $y^2=2(x-3)$ at which the normal is parallel to the line $y-2x+1=0$ is

The equation of the diameter of the parabola $y^2 = x$ corresponding to the chord $x - y + 1 = 0$ is