Question

Pq is the double ordinate of parabola y^2 = 4ax. The locus of the points of trisection of pq is

Answer 1

Given: PQ is the double ordinate of parabola y^2 = 4ax.

To find: The locus of the points of trisection of PQ.

Solution:

• So. first let the points of P be (l,m). This P trisect the double ordinate LL′

•  Let the coordinates of L be (at², 2at) and L′  be (at²,−2at)

• Now, the coordinates of point which divides LL′  in 1:2 are  

              P( { at²(1) + at²(2) }  / 2+1,  { -2at(1) +2at(2) } / 2+1

             P(at² ,  2at /3)

• Now lets compare the points, we get:

             ​l=at ² ......(i)          and     m=  2at /3 ........(ii)

             ​from (ii), we can get:

             t = 3m/2a  

• Now, substitute the value of t in (i) , we get:

            m=a(  3m  / 2a)²

            m=a  (9m²/4a²)

            9m² = 4al

• Replacing l by x and m by y

             9y² = 4ax

Answer:

               Hence the locus is y² = (4/9) ax.