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 \(Q\), ...

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 cont...

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...

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 \(h>2\).