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:

Find the equation of the parabola whose axis is parallel to the $y$-axis and which passes through the points $(0,4), (1,9)$ and $(4,5)$.

The length of the chord of the parabola $x^2 = 4ay$ passing through its vertex and having slope $\tan\alpha$ is:

If tangent lines are drawn from the point $(-1, 2)$ to the parabola $y^2 = 4x$, then the area of the triangle (in sq. units) formed by the chord of contact and the tangents drawn is: (in $\sqrt{2}$)

The line $y = mx + 1$ is a tangent to the parabola $y^2 = 4x$. Find the value of $m$.

Let $L_1, L_2$ be the lines passing through the point $P(0,1)$ and touching the parabola $9x^2+12x+18y-14=0$. Let $Q$ and $R$ be the points on the lines $L_1$ and $L_2$ such that the $\triangle PQR$ is an isosceles triangle with base $QR$. If the slopes of the lines $QR$ are $m_1$ and $m_2$,then $16(m_1^2+m_2^2)$ is equal to ..............