Question

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.

Solution

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

                           

    

SIMILAR QUESTIONS

Q1

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Q2

x and y are the sides of two squares such that y = x – x2. Find the rate of change of the area of the second square with respect to the first square.

Q3

Find equation of tangent to the curve 2y = x2 + 3 at (x1y1).

Q4

Find the equation of tangent to the curve y2 = 4ax at (at2, 2at).

Q5

Find the sum of the intercepts on the axes of coordinates by any tangent to the curve,  

                                

Q6

The tangent represented by the graph of the function y = (x) at the point with abscissa x = 1 form an angle π/6 and at the point x = 2 an angle of π/3 and at the point x = 3 an angle π/4. Then find the value of,  

                                                

Q7

Find the acute angle between the curves y = | x2 – 1| and y = | x2 – 3 | at their points of intersection when x > 0.

Q8

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Q9

 in which interval

Q10

If f (x) = xα log x and f (0) = 0, then the value of ‘α’ for which Rolle’s theorem can be applied in [0, 1] is: