Question

In the standard (x,y) coordinate plane, the center of the circle shown below lies on the x-axis at x=4. If the circle is tangent to the y-axis, which of the following is an equation of the circle?

1) x² + y² = 16
2) (x-4)² + y² = 16
3) x² + (y-4)² = 16
4) (x-4)² + (y-4)² = 16

Answer 1

The correct equation of the circle is **(x-4)² + y² = 16** option 2.

Step-by-step explanation:

In the given problem, we are told that the center of the circle lies on the x-axis at x=4. This implies that the y-coordinate of the center is 0. Furthermore, since the circle is tangent to the y-axis, the distance from the center (x=4, y=0) to the y-axis is equal to the radius of the circle.

Now, the distance from a point (x, y) to the y-axis is given by the absolute value of x. Therefore, the radius of the circle is |4 - 0| = 4.

Now, the general equation of a circle with center (h, k) and radius r is given by (x-h)² + (y-k)² = r². Plugging in the values we have, the equation becomes (x-4)² + (y-0)² = 4², which simplifies to (x-4)² + y² = 16. Thus, the correct answer is option 2.

In conclusion, by utilizing the information about the center of the circle and its tangency to the y-axis, we can deduce the correct equation as (x-4)² + y² = 16. This demonstrates the application of fundamental concepts in coordinate geometry and circle equations to solve the given problem accurately option 2.