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

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

What is the focus of the parabola $y^2 - x - 2y + 2 = 0$?

Find the vertex,focus,axis,length of latus rectum,and the equation of the directrix for the parabola $y^{2} = 4x + 4y$.

An equilateral triangle is inscribed in the parabola $y^2=16ax$ with one of its vertices at the origin. Then,the centroid of that triangle 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