The vertex of the parabola $y^2 + 2y + x = 0$ lies in which quadrant?

The normals at three points $P, Q, R$ of the parabola $y^2 = 4ax$ meet in $(h, k).$ The centroid of triangle $PQR$ lies on :

The normal at a point on the parabola $y^2 = 4x$ passes through a point $P$. Two more normals to this parabola also pass through $P$. If the centroid of the triangle formed by the feet of these three normals is $G(2,0)$,then the abscissa of $P$ is

The Cartesian coordinates of the point on the parabola $y^{2}=x$ whose parameter is $t = -\frac{4}{3}$ are

The length of the latus rectum of $3x^{2} - 4y + 6x - 3 = 0$ is

Let $P(x, y)$ be a variable point on the parabola $y = 4x^2 + 1$. Let $Q(c, c)$ be the foot of the perpendicular drawn from $P$ to the line $y = x$. If $R(h, k)$ is the mid-point of $PQ$,then the locus of $R$ is: