Question

Find the locus of the mid-point of the chords of the parabola y2 = 4ax which subtend a right angle at the vertex of the parabola.

Solution

Correct option is

y2 – 2ax + 8a2 = 0

Let P(h, k) be the mid-point of a chord QR of the parabola

y2 = 4ax, then equation of chord QR is

               T = S1

or     yk – 2a(x + h) = k2 – 4ah

     yk – 2ax = k2 – 2ah

Let A is vertex of the parabola. For combined equation of AQand AR, use homogenization of y2 = 4ax with the help of (1).

  

    y2(k2 – 2ah) – 4akxy +8a2x2 = 0

Since 

 Coefficient of x2 + coefficient of y2 = 0

        k2 – 2ah + 8a2 = 0

Hence the locus of P(h, k) is

          y2 – 2ax + 8a2 = 0

 

 

SIMILAR QUESTIONS

Q1

The equation of the latus rectum of the parabola x2 + 4x + 2= 0 is 

Q2

x – 2 = t2y = 2t are the parameter equations of the parabola 

Q3

The equation  represents a parabola if  is

Q4

t1’ and ‘t2’ are two points on the parabola y2 = 4x. If the chord joining them is a normal to the parabola at ‘t1’ then

Q5

The vertex of the parabola y2 = 8x is at the center of a circle and the parabola cuts the circle at the ends of its latus rectum. Then the equation of the circle is

Q6

Find the equation of the parabola whose focus is (1, 1) and the directrix is x + y + 1 = 0.

Q7

If the line 2x + 3y = 1 touch the parabola y2 = 4ax at the pointP. Find the focal distance of the point P.

Q8

Find the angle between the tangents of the parabola y2 = 8x, which are drawn from the point (2, 5).

Q9

Find the locus of middle point of chord y2 = 4ax drawn through vertex.

Q10

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