Question

If (r = (x, y) mid x² + y² = a²) is a relation, then its inverse (r' = (x, y) mid y² + x² = a²) would be the same as (r). Why is that? Can someone please explain it with detail?

a) The properties of circles
b) The symmetry of the relation
c) The commutative property of addition
d) The associative property of multiplication

Answer 1

The inverse of the relation x² + y² = a² is y² + x² = a², and both equations represent the same circle.

Step-by-step explanation:

In this case, the relationship represented by the equation x² + y² = a² is that of a circle with radius 'a' centered at the origin. The equation y² + x² = a² represents the same circle, but with the x and y coordinates swapped. This is because the equation is symmetric with respect to the x and y axes.

For example, if we take a point (x, y) that lies on the original circle, its coordinates (y, x) will lie on the inverse circle. This is because the distance of any point on the circle from the origin remains the same, regardless of the order of the x and y coordinates.

Therefore, the inverse of the relation x² + y² = a² is y² + x² = a², and both equations represent the same circle.