Find The Shortest Distance Between The Circle x2 + y2 – 24y + 128 = 0 And The Parabola y2 = 4x.

Show that the normal at a point (at², 2at) on the parabola y² = 2ax cuts the curve again at the point whose parameter .

Find the locus of a pint P which moves such that two of the three normal’s drawn from it to the parabola y² = 4ax are mutually perpendicular.

If normal at the point (at², 2at) in the parabola y² = 4axintersects the parabola again at the (am², 2am), then find the minimum value of m².

The equation of circle touching the parabola y² = 4x at the point  (1, –2) and passing through origin is

The vertex of a parabola is the point (a, b) and latus-rectum is of length l. If the axis of the parabola is along the positive direction of y-axis. Then its equation is

Slope of common tangent to parabolas y² = 4x and x² = 8y is

If a focal chord with positive slope of the parabola y² = 16xtouches the circle x² + y² – 12x + 34 = 0, then m is

If 2y = x + 24 is a tangent to parabola y² = 24x, then its distance from parallel normal is

PQ is a focal chord of the parabola y² = 4ax, O is the origin. Find the coordinates of the centroid, G, of triangle OPQ and hence find the locus of G as PQ varies.

The equation of the directrix of the parabola y² + 4y + 4x + 2 = 0 is