Three normals are drawn from the point (c, 0) to the curve y2 = x, show that c must be greater than ½. One normal is always the x-axis. Find c for which the other normals are perpendicular to each other.


Correct option is

c > 1/2 ,        c = 3/4

The slope form of the normal to the curve y2 = 4ax is,    

                           y = mx – 2am – am3                                   …(i)  

For the curve given y2 = x, we have   

                           4a = 1 ⇒ a = 1/4  

∴ Equation of normal is,   


The equation passes through (c, 0) then,   




For m = 0, the normal is y = 0 which is the x-axis.    

The other two values of m are given by,  



If c = 1/2, then m = 0 which is already considered  

So,                                     c > 1/2  

Now, for the other two normals to be perpendicular to each other, we must have





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