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show that the locus of the points of trisection of the double ordinates of a parabola forms a parabola

show that the locus of the  points of trisection of the double ordinates of  a parabola forms a parabola

Grade:11

1 Answers

Arun
25763 Points
3 years ago
Dear student
 
Let y2 = 4ax be the parabola. Let AB be the Latus rectum.
Let CD be an arbitrary double ordinate line parallel to the latus rectum (Points C and D are on the parabola) and let points M and N be the points of trisection i.e. they divide the double ordinate CD in 3 equal parts.
Our objective is find the locus or path of these points M and N. Simply by observation, we can say that the path would be a parabola.
We will now proceed to find the equation this parabola i.e. the locus of M and N.
Let the co-ordinates of point C be (x, y). Pont D would be (x, -y).
Let the co-ordinates of M be (m,n) and co-ordinates of point N be (m, -n)
Now m = x
And n = (-y+2y)/3 = y/3 or y=3n
Now putting the values of m and n into the equation of the parabola, we get
(3n)2 = 4am or n2 = (4/9)am
And this is the equation of the parabola.
 
Regards
Arun (askIITians forum expert)

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