The equation of the parabola whose focus is $(5, 3)$ and directrix is $3x - 4y + 1 = 0$ is:

The tangent $PT$ and the normal $PN$ to the parabola $y^2=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$,respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose
$(A)$ vertex is $\left(\frac{2a}{3}, 0\right)$
$(B)$ directrix is $x=0$
$(C)$ latus rectum is $\frac{2a}{3}$
$(D)$ focus is $(a, 0)$

Find the equation of the tangent to the curve $y^2 = 6x$ at the point $(2, -3)$.

If a point $P$ moves such that its distances from the point $A(1, 1)$ and the line $x+y+2=0$ are equal,then the locus of $P$ is

Find the parameter $t$ for the point $(4, -6)$ on the parabola $y^2 = 9x$.

Consider the parabola $y^{2}=4x$. Let $P$ and $Q$ be points on the parabola where $P(4, -4)$ and $Q(9, 6)$. Let $R$ be a point on the arc of the parabola between $P$ and $Q$. Then,the area of $\Delta PQR$ is largest when