The point on the parabola $y^{2}=64x$ which is nearest to the line $4x+3y+35=0$ has coordinates

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

If the chord joining the points $P_{1}(x_{1}, y_{1})$ and $P_{2}(x_{2}, y_{2})$ on the parabola $y^{2} = 12x$ subtends a right angle at the vertex of the parabola,then $x_{1}x_{2} - y_{1}y_{2}$ is equal to

The vertices of the base of an isosceles triangle lie on a parabola $y^2=4x$ and the base is a part of the line $y=2x-4$. If the third vertex of the triangle lies on the $X$-axis,its coordinates are

Let $PQ$ be a chord of the parabola $y^2=12x$ and the midpoint of $PQ$ be at $(4,1)$. Then,which of the following points lies on the line passing through the points $P$ and $Q$?

Let the point $P$ of the focal chord $PQ$ of the parabola $y^2=16x$ be $(1, -4)$. If the focus of the parabola divides the chord $PQ$ in the ratio $m:n$,where $\operatorname{gcd}(m, n)=1$,then $m^2+n^2$ is equal to: