The coordinates of the focus of the parabola described parametrically by $x=5t^2+2, y=10t+4$ (where $t$ is a parameter) are

For the parabola $y^2 = 8x$,let $\Delta_1$ be the area of the triangle formed by the endpoints of the latus rectum and the point $P \left( \frac{1}{2}, 2 \right)$ on the parabola. Let $\Delta_2$ be the area of the triangle formed by the tangents at the endpoints of the latus rectum and the tangent at point $P$. Find the value of $\frac{\Delta_1}{\Delta_2}$.

The focus of the parabola $4y^2 - 6x - 4y = 5$ is

The acute angle between the tangents drawn at the point of intersection (other than the origin) of the curves $x^2=4y$ and $y^2=4x$ is

The shortest distance between the line $y = x$ and the curve $y^2 = x - 2$ is

The line $y=x+1$ is a tangent to the curve $y^{2}=4x$ at the point