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

The locus of the foot of the perpendicular drawn from the focus to any tangent of the parabola $y^2 = 4ax$ is:

The equations of two sides of a variable triangle are $x = 0$ and $y = 3$,and its third side is a tangent to the parabola $y^2 = 6x$. The locus of its circumcentre is:

The tangent drawn at any point $P$ to the parabola ${y^2} = 4ax$ meets the directrix at the point $K$. Then the angle which $KP$ subtends at its focus is ............. $^\circ$.

The equation of the tangent to the parabola $y^{2}=4x$ inclined at an angle of $\frac{\pi}{4}$ to the positive direction of $x$-axis is:

Let $\alpha_1$ and $\alpha_2$ be the ordinates of two points $A$ and $B$ on a parabola $y^2=4ax$ and let $\alpha_3$ be the ordinate of the point of intersection of its tangents at $A$ and $B$. Then,$\alpha_3-\alpha_2=$