Find, what the following equations become when the origin is shifted to the point (1, 1).
(i) x2 + xy – 3y2 – y + 2 = 0
(ii) xy – y2 – x + y = 0
(iii) xy – x – y + 1 = 0
(iv) x2 – y2 – 2x + 2y = 0
To find: Transformed equations of given equations when the origin (0, 0) is shifted at point (1, 1).
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).
(i) x2 + xy – 3y2 – y + 2 = 0
Substituting x and y with (x+1) and (y+1) respectively, we have
= (x+1)2 + (x+1)(y+1) - 3y2 – (y + 1) + 2 = 0
= x2 + 1 + 2x + xy + x + y +1 – 3y2 – y – 1 + 2 = 0
= x2 - 3y2 + xy +3x – 6y = 0
Hence, the transformed equation is x2 - 3y2 + xy +3x – 6y = 0
(ii) xy – y2 – x + y = 0
Substituting x and y with (x+1) and (y+1) respectively, we have
= (x+1)(y+1) - y2 – (x + 1) + (y + 1) = 0
= xy + x + y +1 – y2 – x – 1 – y – 1 = 0
= xy - y2 = 0
Hence, the transformed equation is xy - y2 = 0
(iii) xy – x – y + 1 = 0
Substituting x and y with (x+1) and (y+1) respectively, we have
= (x+1)(y+1) - (x + 1) – (y + 1) + 1 = 0
= xy + x + y +1 – y – 1 – x – 1 + 1 = 0
= xy = 0
Hence, the transformed equation is xy = 0.
(iv) x2 – y2 – 2x + 2y = 0
Substituting x and y with (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.