Parabola

The equation of the ellipse having foci (2, 0), ( -2, 0) and minor axis of length 8 units is:

x=t²+ 1 & y = 2t +1. Tha cartesian equation of its directrix is

Equation of the normal at t is

tx + y = 2at + at³

The foci of a hyperbola coincide with the foci of the ellipse \(\rm \frac{x^2}{25}+\frac{y^2}{16}=1\), find the equation of the hyperbola if its eccentricity is 3.

Find the coordinates of the point whose chord of contact with respect to the parabola y² + 4x is 2x – 7y + 2 = 0

The locus of the point of intersection of perpendicular tangents to a parabola is the directrix of the parabola. This locus is called director circle even though it is a line.

Suppose the normals at three different points on the parabola y² = 4x pass through the point (h, 0). Show that h > 2

If two tangents to a parabola make complementary angles with the axis, then show that their point of intersection lies on the line x = a.

If 2x + y + k = 0 is a normal to the parabola y² = − 8x, then the value of k is

(A)  12

(B)   24

(C)  − 12

(D)  − 24