The equation of image of curve ay2 = x3 with respect to the line x + y = a  is:

Asked by yeeshuraj3 | 25th Jul, 2016, 04:04: PM

Expert Answer:

Let space the space equation space of space straight a space line space is space straight a subscript 1 straight x space plus space straight b subscript 1 straight y space plus space straight c space equals space 0 space and space image space of space straight a space point space straight P left parenthesis straight x subscript 1 comma space straight y subscript 1 right parenthesis space is space straight Q left parenthesis straight x subscript 2 comma space straight y subscript 2 right parenthesis space is space gven space by comma
fraction numerator straight x subscript 2 minus straight x subscript 1 over denominator straight a subscript 1 end fraction equals fraction numerator straight y subscript 2 minus straight y subscript 1 over denominator straight b subscript 1 end fraction equals negative fraction numerator 2 open parentheses straight a subscript 1 straight x subscript 1 space plus space straight b subscript 1 straight y subscript 1 space plus space straight c space close parentheses over denominator straight a subscript 1 squared plus straight b subscript 1 squared end fraction

Given space equation space of space the space curve space is space ay squared space equals space straight x cubed

Three space points space on space the space curve space are space left parenthesis 0 comma space 0 space right parenthesis comma space left parenthesis straight a comma space straight a right parenthesis space space space space and space space left parenthesis straight a comma space minus straight a right parenthesis.


Image space of space the space point space space left parenthesis 0 comma space 0 right parenthesis space with space respect space to space line space straight x space plus space straight y space equals space straight a space is space given space by

fraction numerator straight x subscript 2 minus 0 over denominator 1 end fraction equals fraction numerator straight y subscript 2 minus 0 over denominator 1 end fraction equals negative fraction numerator 2 open parentheses 0 space plus space 0 space minus straight a space close parentheses over denominator 1 squared plus 1 squared end fraction
straight x subscript 2 space equals space straight a comma space space straight y subscript 2 space space equals space straight a

therefore Image space of space the space point space space left parenthesis 0 comma space 0 right parenthesis space with space respect space to space line space straight x space plus space straight y space equals space straight a space is space left parenthesis straight a comma space straight a right parenthesis.

Simillarly comma
Image space of space the space point space space left parenthesis straight a comma space straight a right parenthesis space with space respect space to space line space straight x space plus space straight y space equals space straight a space is space left parenthesis 0 comma space 0 right parenthesis.
Image space of space the space point space space left parenthesis straight a comma space minus straight a right parenthesis space with space respect space to space line space straight x space plus space straight y space equals space straight a space is space left parenthesis 2 straight a comma space 0 right parenthesis.

Now space we space have space three space point space left parenthesis straight a comma space straight a right parenthesis comma space left parenthesis 0 comma space 0 right parenthesis space space and space left parenthesis 2 straight a comma space 0 right parenthesis space lying space on space the space image space curve.

Finally space substitute space these space points space in space the space standard space equation space of space the space curve space passing space through space three space point space.

Answered by Vijaykumar Wani | 27th Jul, 2016, 04:58: PM