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.
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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