Answer :


To find: Transformed equation of given equation when the origin (0, 0) is shifted at point (ab/(a – b), 0).


We know that, when we transform origin from (0, 0) to an arbitrary point (p, q), the new coordinates for the point (x, y) becomes (x + p, y + q), and hence an equation with two variables x and y must be transformed accordingly replacing x with x + p, and y with y + q in original equation.


Since, origin has been shifted from (0, 0) to (1, 1); therefore any arbitrary point (x, y) will also be converted as (x + 1, y + 1) or (x + 1, y + 1).


(i) x2 + xy – 3x – y + 2 = 0


Substituting the value of x by x + 1 and y by y + 1, we have


= (x + 1)2 + (x + 1)(y + 1) – 3(x + 1) – (y + 1) + 2 = 0


= x2 + 1 + 2x + xy + x + y + 1 – 3x – 3 - y - 1 + 2 = 0


= x2 + xy = 0


Hence, the transformed equation is x2 + xy = 0.


(ii) x2 – y2 – 2x + 2y = 0


Substituting the value of x and y by x + 1 and y + 1 respectively, we have


= (x + 1)2 – (y + 1)2 – 2(x + 1) + 2(y + 1) = 0


= x2 + 1 + 2x - y2 – 1 – 2y – 2x – 2 + 2y + 2 = 0


= x2 - y2 = 0


Hence, the transformed equation is x2 - y2 = 0.


(iii) xy – x – y + 1 = 0


Substituting the value of x and y by x + 1 and y + 1 respectively, we have


= (x + 1)(y + 1) – (x + 1) - (y + 1) + 1 = 0


= xy + x + y + 1 – x – 1 – y – 1 + 1 = 0


= xy = 0


Hence, the transformed equation is xy = 0.


(iv) xy – y2 – x + y = 0


Substituting the value of x and y by x + 1 and y + 1 respectively, we have


= (x + 1)(y + 1) – (y + 1)2 - (x + 1) + (y + 1) = 0


= xy + x + y + 1 – y2 – 1 – 2y - x – 1 + y + 1 = 0


= xy - y2 = 0


Hence, the transformed equation is xy - y2 = 0.


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