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

What is the focus of the parabola $y^2 - x - 2y + 2 = 0$?

The normal at a point on the parabola $y^2=4x$ passes through $(5,0)$. If there are two more normals to this parabola passing through $(5,0)$,then the equation of one of these normals is

Let $S$ be the set of all $a \in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $P(b, c)$,where $b, c \in \mathbb{N}$,on the parabola $y^2 = 2ax$ and the lines $x = b$ and $y = 0$ is $16 \text{ unit}^2$. Then $\sum_{a \in S} a$ is equal to $..........$.

If the normal drawn from the point $(t_1^2, 2t_1)$ to the parabola $y^2 = 4x$ intersects the parabola again at the point $(t_2^2, 2t_2)$,then -

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