Question

If the straight line p = xcos α + ysin α touch the curve (x/a)^m + (y/b)^m = 1, prove that 

.

Answer 1

The given equation of the curve is 

                   ......(1)

On differentiating with respect to x, we get 

or Slope of the tangent at (x,y) = dy/dx = -(b/a)^m (x/y)^{m - 1}         

Equation of the tangent at (x, y) becomes

or 

 [using   (1)]                   ......(2)

Now if the line x cos α + y sin α = p touch the given curve then it should be identical to equation (2)

Whence 

 

=

 

therefore

= 1

whence the result.