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if the cord joining the points t1 and t2 to the parabola y^2 =4ax is normal to the parabola at t1 then prove that t1(t1+t2)=-2

if the cord joining the points t1 and t2 to the parabola y^2 =4ax is normal to the parabola at t1 then prove that t1(t1+t2)=-2

Grade:11

1 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Rohit,

 

y2 = 4ax, will give dy/dx = 2a/y.

Hence, slope of tangent at t1 is 2a/2at1 = 1/t1.

So slope of normal at point t1 is -t1.

 

Also slope of line joining t1 and t2 is (2at2 - 2at1 / at22 - at12), which is equal to 2/(t1+t2).

 

The two slopes are the same. Hence

2/(t1+t2) = -t1, or t1(t1+t2) = -2.

That solves the problem.

 

Hope that helps.

 

All the best,

Regards,

Ashwin (IIT Madras)

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