Question

Find the point on the curve y = x2 which is closest to the point A(0, a).  

Solution

Correct option is

Using the parametric representation, consider an arbitrary point P (tt2) on the curve.

Distance of P from A = PA   

                  

We have to find t so that this distance is minimum.

We will minimize PA2  

Let         PA2 = (t) = t2 + (t2 – a)2   

           

             

             

             

We have to consider two possibilities.   

Case I: a = 1/2   

In this case, t = 0 is the only value.  

            

Hence the closest point corresponds to t = 0

⇒        (0, 0) is the closest point.   

Case II: a > ½

            

            

            

                                      

Hence the distance is minimum for   

So the closest points are

                    

SIMILAR QUESTIONS

Q1

Find points of local maximum and local minimum of (x) = x2/3 (2x – 1). 

Q2

Identify the absolute extrema for the following function.  

                      f (x) = x2 on [–1, 2]

Q3

Identify the absolute extrema for the following function.  

                     f (x) = x3   or    [–2, 2]

Q4

Determine the absolute extrema for the following function and interval.

             g(t) = 2t3 + 3t2 – 12t + 4    on    [– 4, 2]  

Q5

Find the local maximum and local minimum values of the function y = xx.

Q6

A window is in the form of a rectangle surmounted by a semi-circle. The total area of window is fixed. What should be the ratio of the areas of the semi-circular part and the rectangular part so that the total perimeter is minimum?   

Q7

A box of constant volume C is to be twice as long as it is wide. The cost per unit area of the material on the top and four sides is three times the cost for bottom. What are the most economical dimensions of the box?  

Q8

Find the maximum surface area of a cylinder that can be inscribed in a given sphere of radius R 

Q9

Find the semi-vertical angle of the cone of maximum curved surface area that can be inscribed in a given sphere of radius R.

Q10

Find the shortest distance between the line y – x = 1 and the curve x = y2.