If the straight line $x + y = 1$ touches the parabola $y^2 - y + x = 0$,then the coordinates of the point of contact are

If the line $x = my + k$ touches the parabola $x^2 = 4ay$,then $k = $

Let $y^2=12x$ be a parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ)=\frac{147}{4}$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of circle $C$ is $64x^2+64y^2-\alpha x-64\sqrt{3}y=\beta$,then $\beta-\alpha$ is equal to . . . . . .

Two parabolas $y^2 = 4a(x - l_1)$ and $x^2 = 4a(y - l_2)$ always touch one another,where $l_1$ and $l_2$ are variable. The locus of their point of contact has the equation:

If the vertex of the conic $y^{2}-4y=4x-4a$ always lies between the straight lines $x+y=3$ and $2x+2y-1=0$, then:

Let $P(\alpha, \beta)$ be a point on the parabola $y^2 = 4x$ which is at minimum distance from the circle $x^2 + y^2 - 4x - 20y + 103 = 0$. Then $\alpha \beta$ is