Question

Given the equation (x-2)^2+y^2=9, determine what it is symmetric with respect to. a) x-axis

b) y-axis
c) y-axis and x = 2
d) x = 2

Answer 1

The circle represented by the equation (x-2)^2+y^2=9 has symmetry with respect to the x-axis and the line x = 2.

Step-by-step explanation:

The equation (x-2)^2+y^2=9 represents a circle with a radius of 3 centered at the point (2,0). To determine its symmetry, we can perform tests by replacing variables to see if the equation remains unchanged.

• x-axis symmetry: Replace y with -y. The equation becomes (x-2)^2+(-y)^2=9, which simplifies to the original equation. Therefore, the circle is symmetric with respect to the x-axis.
• y-axis symmetry: Replace x with -x. The equation becomes (-x-2)^2+y^2=9, which does not simplify to the original equation. Therefore, the circle is not symmetric with respect to the y-axis.
• Symmetry with respect to x=2: Replace x with 2+(2-x). The equation remains unchanged as (2+(2-x)-2)^2+y^2=9 simplifies to the original equation. Hence, the circle is also symmetric with respect to the line x=2.

Based on these tests, the circle given by the equation (x-2)^2+y^2=9 is symmetric with respect to both the x-axis and the line x=2, so the correct answer is (c) y-axis and x = 2.