The equation of the tangent to the parabola $y^2 = 4ax$ at the point $(3, 2)$ is . . . .

The ordinates of the points $P$ and $Q$ on the parabola with focus $(3,0)$ and directrix $x = -3$ are in the ratio $3:1$. If $R(\alpha, \beta)$ is the point of intersection of the tangents to the parabola at $P$ and $Q$,then $\frac{\beta^2}{\alpha}$ is equal to $.............$.

The maximum area of a circle centered at the origin,which is inscribed in the parabola $y = x^2 - 100$,can be expressed as $\frac{a\pi}{b}$,where $a$ and $b$ are coprime numbers. Then the value of $a + b$ is:

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

For any non-zero real value of $m$,the equation of the parabola to which the line $m x-y+10+m^2=0$ is a tangent,is

If a chord of the parabola $y^2=4x$ passes through its focus and makes an angle $\theta$ with the $X$-axis,then its length is