The condition for which the straight line $y = mx + c$ touches the parabola $y^2 = 4ax$ is

If $S(a, b)$ is a fixed point and $P(\alpha, \beta)$ is a variable point such that $4[(x-a)^2+(y-b)^2]=(\alpha x+\beta y+7)^2$ represents a parabola,then the locus of $P(\alpha, \beta)$ is

The sum of squares of all possible values of $k$,for which the area of the region bounded by the parabolas $2y^2 = kx$ and $ky^2 = 2(y - x)$ is maximum,is equal to:

The points $(at_1^2, 2at_1)$,$(at_2^2, 2at_2)$,and $(a, 0)$ will be collinear,if

The line $x - y + 2 = 0$ touches the parabola $y^2 = 8x$ at the point

If one of the vertices of an equilateral triangle inscribed in the parabola $y^2=12x$ coincides with the vertex of the parabola, then the area (in sq. units) of that triangle is (in $\sqrt{3}$)