Let the parabola $y = x^2 + px + q$ passing through the point $(1, -1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2 + q^2$ is :

Let A be the focus of the parabola $y^2 = 8x$. Let the line $y = mx + c$ intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is $\left( \frac{7}{3}, \frac{4}{3} \right)$, then $(BC)^2$ is equal to :

Let the image of parabola $x^2=4 y$, in the line $x-y=1$ be $(y+a)^2=b(x-c)$, $a, b, c \in \mathrm{~N}$. Then $a+b+c$ is equal to

An equilateral triangle OAB is inscribed in the parabola $y^2=4 x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having $A B$ as a diameter from the origin is