If $2x + 3y + 12 = 0$ and $x - y + 4\lambda = 0$ are conjugate with respect to the parabola $y^2 = 8x$,then $\lambda$ is equal to

The equation of the tangent to the parabola $y = x^2 - x$ at the point where $x = 1$ is:

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:

Statement $I$: $4x^2+y^2-4xy-30x-50y+40=0$ is the equation of a parabola having $(2,3)$ as its focus and $x+2y+5=0$ as its directrix.
Statement $II$: The equation of the directrix of the parabola $x^2-4x+16y+52=0$ is $y+1=0$.
Which of the above statements is (are) true?

Let $P_{1}$ be a parabola with vertex $(3,2)$ and focus $(4,4)$,and let $P_{2}$ be its mirror image with respect to the line $x + 2y = 6$. Then the directrix of $P_{2}$ is $x + 2y =$

Find the coordinates of the focus,axis of the parabola,the equation of the directrix,and the length of the latus rectum for $x^{2} = -16y$.