Find the equation of the chord of the parabola $y^2 = 6x$ passing through the vertex and the negative end of the latus rectum.

Let $(x, y)$ be any point on the parabola $y^2 = 4x$. Let $P$ be a point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $1 : 3$. Find the locus of $P$.

Find the equation of the parabola that satisfies the following conditions: Vertex $(0, 0)$,focus $(-2, 0)$.

Let a circle tangent to the directrix of a parabola $y^2 = 2ax$ have its center coinciding with the focus of the parabola. Then the points of intersection of the parabola and the circle are

Let a parabola $P$ be such that its vertex and focus lie on the positive $x$-axis at a distance $2$ and $4$ units from the origin,respectively. If tangents are drawn from $O(0,0)$ to the parabola $P$ which meet $P$ at $S$ and $R$,then the area (in $sq. \text{ units}$) of $\triangle SOR$ is equal to:

Find the equation of the common tangent to the parabolas $y = x^2$ and $y = -(x - 2)^2$.