Q. The Straight line  Y=tX+2a+a Cut the Parabola  =4aX  at  A and B. If  AB subtends a Right angle at origin ,find gradient of AB?

If we homogenise this equation we get the combined equation of AO and OB.So doing that we get:

We write 'a' as a = y-tx/2t^2 + t^3. Putting this in the equation of parabola: 

 y^2 = 4(yx - tx^2)/2t^2 + t^3

 Now coeffecient of (x^2) + coeffecient of (y^2) =0

  => 2t^2 + t^3 +4t =0 => t^2 + 2t +4 =0 = > (t+1)^2 =0 = > t=-1

And from the equation of straight line we find the slope is t = -1