Parabola

The equation of the directrix of the parabola ${y^2} + 4y + 4x + 2 = 0$

If the line $x - 1 = 0$ is the directrix of the parabola ${y^2} - kx + 8 = 0,$ then one of the values of $k$ is

If $x + y = k$ is normal to ${y^2} = 12x,$ then $k$ is

Let ${C_1}$ and ${C_2}$ be respectively, the parabolas ${x^2} = y - 1$ and ${y^2} = x - 1$. Let $P$ be any point on ${C_1}$ and $Q$ be any point on ${C_2}$. Let ${P_1}$ and ${Q_1}$ be the reflections of $P$ and $Q$, respectively, with respect to the line $y=x$. Prove that ${P_1}$ lies on ${C_2}$, ${Q_1}$ lies on ${C_1}$ and $PQ \ge $ min $\left\{ {P{P_1},Q{Q_1}} \right\}$. Hence or otherwise determine points ${P_0}$ and ${Q_0}$ on the parabolas ${C_1}$ and ${C_2}$ respectively such that ${P_0}{Q_0} \le PQ$ for all pairs of points $(P,Q)$ with $P$ on ${C_1}$ and $Q$ on ${C_2}$.

The curve described parametrically by $x = {t^2} + t + 1,$ $y = {t^2} - t + 1 $ represents

Points $A, B$ and $C$ lie on the parabola ${y^2} = 4ax$. The tangents to the parabola at $A, B$ and $C$, taken in pairs, intersect at points $P, Q$ and $R$. Determine the ratio of the areas of the triangles $ABC$ and $PQR$.

From a point $A$ common tangents are drawn to the circle ${x^2} + {y^2} = {a^2}/2$ and parabola ${y^2} = 4ax$. Find the area of the quadrilateral formed by the common tangents, the chord of contact of the circle and the chord of contact of the parabola.

Consider a circle with its centre lying on the focus of the parabola ${y^2} = 2px$ such that it touches the directrix of the parabola. Then a point of intersection of the circle and parabola is

Show that the locus of a point that divides a chord of slope $2$ of the parabola ${y^2} = 4x$ internally in the ratio $1:2$ is a parabola. Find the vertex of this parabola.

Through the vertex $O$ of parabola ${y^2} = 4x$, chords $OP$ and $OQ$ are drawn at right angles to one another . Show that for all positions of $P$, $PQ$ cuts the axis of the parabola at a fixed point. Also find the locus of the middle point of $PQ$.

The point of intersection of the tangents at the ends of the latus rectum of the parabola ${y^2} = 4x$ is ...... .

Three normals are drawn from the point $(c, 0)$ to the curve ${y^2} = x.$ Show that $c$ must be greater than $1/2$. One normal is always the $x$-axis. Find $c$ for which the other two normals are perpendicular to each other.

$A$ is point on the parabola ${y^2} = 4ax$. The normal at $A$ cuts the parabola again at point $B$. If $AB$ subtends a right angle at the vertex of the parabola. Find the slope of $AB$.

Suppose that the normals drawn at three different points on the parabola ${y^2} = 4x$ pass through the point $(h, k)$. Show that $h>2$.