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  • The beds of two rivers (within a certain region) are a parabola y =  and a straight line y = x - 2. These rivers are to b

The beds of two rivers (within a certain region) are a parabola y = x^2 and a straight line y = x - 2. These rivers are to be connected by a straight canal. The coordinates of the ends of the shortest canal can be:

Option: 1

( 1/2 , 1/4 ) and ( -11/8 , 5/8 )


Option: 2

( 1/2 , 1/4 ) and ( 11/8 , -5/8 )


Option: 3

(0,0) and (1,-1)


Option: 4

none 


Answers (1)

best_answer

 

Equation of Normal -

Equation of normal to the curve  y = f(x) at the point  P(x1, y1) on the curve having a slope  MN  is 

(y-y_{1})=M_{N}(x-x_{1})


=\frac{-1}{\frac{dy}{dx}_{(x_{1},y_{1})}}(x-x_{1})

-

 

 

        y = x^2                                                                                                                                  

            dy/dx = 2x

            2x = 1    

\Rightarrow x = 1/2

            (1/2,1/4 ) on parabola

            shortest canal will be along the common normal of y = x^2and y = x – 2 which will be,

            y = – x + c

            Q  it passes through (1/2,1/4 )   Þ  c = 3/4

            solving,  y = x – 2 and   y = – x + 3/4

            y = – 5/8 and x = 11/8

            Hence  point on straight line along the shortest canal is (11/8 , -5/8 )

 

Posted by

Deependra Verma

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