Given, equation (x – a²) + (y – b)² = r². For curious readers- this equation represents a circle in the space centered at point (a, b) having a radius of r units.

To find: Transformed equation of given equation when the coordinate axes are transformed parallelly at point (a - c, b).

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

The given equation (x – a)² + (y – b)² = r² will hence be transformed into the new equation by changing x by x – a + c and y by y – b, i.e. substitution of x by x + a and y by y + b.

= ((x + a - c) – a)² + ((y – b ) - b)² = r²

= (x – c)² + y² = r²

= x² + c² – 2cx + y² = r²

= x² + y² = r² - c² + 2cx

Hence, the transformed equation is x² + y² = r² - c² + 2cx.