Question

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

Solution

Correct option is

Let P(at2, 2at). Then Q is (at22, 2at2) with the condition t1t2 = –1

(Condition for focal chord).

 if G(x, y) be the centroid of , then

    

Eliminating t1 and t2 between x and y, we have

     

           (using t1t2 = –1)

 The locus of G(as t1 varies) is 9y2 = 12ax – 8a2

  

Which is a parabola with vertex at  and latus rectum of length .

SIMILAR QUESTIONS

Q1

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

Q2

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

Q3

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.

Q4

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.

Q5

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

Q6

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

Q7

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

Q8

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

Q9

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

Q10

Find the shortest distance between the circle x2 + y2 – 24y + 128 = 0 and the parabola y2 = 4x.