What does the equation (a – b) (x2 + y2) – 2abx = 0 become if the origin is shifted to the point (ab/(a-b), 0) without rotation?

Given, equation (a – b)(x2 + y2) – 2abx = 0


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 (ab/(a – b), 0); therefore any arbitrary point (x, y) will also be converted as (x + (ab / (a - b)), y + 0) or (x + ab / (a - b), y).


The given equation (a – b)(x2 + y2) – 2abx = 0will hence be transformed into new equation by changing x by x + ab/(a-b) and y by y as





Hence, the transformed equation is (a – b)2 (x2 + y2) = a2 b2.


2