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

The focus of the parabola ${x^2} = 2x + 2y$ is

What is the angle between the tangents drawn from $(1, 4)$ to the parabola $y^2 = 4x$?

$A$ point on the parabola whose axis is parallel to the $X$-axis and which passes through the points $(0,1), (3,0), (0,-2)$ is

Let $P$ be a point on the parabola $x^2 = 4y$. If the distance of $P$ from the centre of the circle $x^2 + y^2 + 6x + 8 = 0$ is minimum,then the equation of the tangent to the parabola at $P$ is:

Consider the parabola $y^2=8x$. Let $\Delta_1$ be the area of the triangle formed by the endpoints of its latus rectum and the point $P\left(\frac{1}{2}, 2\right)$ on the parabola,and $\Delta_2$ be the area of the triangle formed by the intersection points of the tangents drawn at $P$ and at the endpoints of the latus rectum. Then $\frac{\Delta_1}{\Delta_2}$ is