The line $y = mx + 1$ is a tangent to the parabola $y^2 = 4x$. Find the value of $m$.

The equation of a tangent to the parabola $y^2 = 8x$ is $y = x + 2$. If another tangent is drawn to the parabola from a point on this line such that it is perpendicular to the given tangent,find the point.

The length of the latus rectum of the parabola ${x^2 - 4x - 8y + 12 = 0}$ is

If the parabola $y = \alpha x^{2} - 6x + \beta$ passes through the point $(0, 2)$ and has its tangent at $x = \frac{3}{2}$ parallel to the $X$-axis,then:

If $x^2 = 8ay$ is the transformed equation of $x^2 - 4y + 6x + 15 = 0$ when the origin is shifted to the point $(\alpha, \beta)$ by translation of axes,then $2\alpha + 8\beta^2 =$

The equation of the tangent to the parabola $y^2=8x$,which is parallel to the line $4x-y+3=0$,is