Find what the following equations become when the origin is shifted to

1.	Find what the following equations become when the origin is shifted to the point (1, 1)?(i) x2 + xy – 3x – y + 2 = 0(ii) x2 – y2 – 2x + 2y = 0(iii) xy – x – y + 1 = 0(iv) xy – y2 – x + y = 0
Answer» (i) x² + xy – 3x – y + 2 = 0 Firstly let us substitute the value of x by x + 1 and y by y + 1 Then, (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 Upon simplification we get, x² + xy = 0 ∴ The transformed equation is x² + xy = 0. (ii) x² – y² – 2x + 2y = 0 Let us substitute the value of x by x + 1 and y by y + 1 Then, (x + 1)² – (y + 1)² – 2(x + 1) + 2(y + 1) = 0 x² + 1 + 2x – y² – 1 – 2y – 2x – 2 + 2y + 2 = 0 Upon simplification we get, x² – y² = 0 ∴ The transformed equation is x² – y² = 0. (iii) xy – x – y + 1 = 0 Let us substitute the value of x by x + 1 and y by y + 1 Then, (x + 1) (y + 1) – (x + 1) – (y + 1) + 1 = 0 xy + x + y + 1 – x – 1 – y – 1 + 1 = 0 Upon simplification we get, xy = 0 ∴ The transformed equation is xy = 0. (iv) xy – y² – x + y = 0 Let us substitute the value of x by x + 1 and y by y + 1 Then, (x + 1) (y + 1) – (y + 1)² – (x + 1) + (y + 1) = 0 xy + x + y + 1 – y² – 1 – 2y – x – 1 + y + 1 = 0 Upon simplification we get, xy – y² = 0 ∴ The transformed equation is xy – y² = 0.

Find what the following equations become when the origin is shifted to the point (1, 1)?(i) x2 + xy – 3x – y + 2 = 0(ii) x2 – y2 – 2x + 2y = 0(iii) xy – x – y + 1 = 0(iv) xy – y2 – x + y = 0

Answer»

(i) x² + xy – 3x – y + 2 = 0

Firstly let us substitute the value of x by x + 1 and y by y + 1

Then,

(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

Upon simplification we get,

x² + xy = 0

∴ The transformed equation is x² + xy = 0.

(ii) x² – y² – 2x + 2y = 0

Let us substitute the value of x by x + 1 and y by y + 1

Then,

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

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

Upon simplification we get,

x² – y² = 0

∴ The transformed equation is x² – y² = 0.

(iii) xy – x – y + 1 = 0

Let us substitute the value of x by x + 1 and y by y + 1

Then,

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

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

Upon simplification we get,

xy = 0

∴ The transformed equation is xy = 0.

(iv) xy – y² – x + y = 0

Let us substitute the value of x by x + 1 and y by y + 1

Then,

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

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

Upon simplification we get,

xy – y² = 0

∴ The transformed equation is xy – y² = 0.