Exploring Circle Intercepts: Where X-Intercepts Equal Y-Intercepts

When dealing with circles in the coordinate plane, it is important to understand the differences and similarities between x-intercepts and y-intercepts. Contrary to a common misconception, x-axis intercepts (x-intercepts) do not necessarily equal y-intercepts on any circle with a center at (h, k). However, there are specific conditions where this equality holds true. In this article, we will explore the concept of circle intercepts and delve into the unique scenario where x-intercepts and y-intercepts are equal.

Understanding Circle Intercepts

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The standard form of the equation of a circle with center at (h, k) and radius r is:

[ (x - h)^2 (y - k)^2 r^2 ]

The x-intercepts of a circle are the points where the circle crosses the x-axis, i.e., when y 0. Similarly, the y-intercepts are the points where the circle crosses the y-axis, i.e., when x 0. Let's examine these intercepts in more detail.

X-Intercepts and Y-Intercepts

To find the x-intercepts of a circle, we set y 0 in the circle's equation:

[ (x - h)^2 (k - k)^2 r^2 Rightarrow (x - h)^2 r^2 ]

Solving for x gives us two possible x-values:

[ x h r ] [ x h - r ]

So, the x-intercepts are (h r, 0) and (h - r, 0).

To find the y-intercepts, we set x 0:

[ (0 - h)^2 (y - k)^2 r^2 Rightarrow h^2 (y - k)^2 r^2 ]

Solving for y gives us two possible y-values:

[ y k sqrt{r^2 - h^2} ] [ y k - sqrt{r^2 - h^2} ]

So, the y-intercepts are (0, k √(r^2 - h^2)) and (0, k - √(r^2 - h^2)).

When X-Intercepts Equal Y-Intercepts

For the x-intercepts to equal the y-intercepts, we need to find the conditions under which the x-intercepts and y-intercepts are the same points. Let's denote the x-intercepts as (a, 0) and (b, 0), and the y-intercepts as (0, c) and (0, d). For these to be equal, we must have:

[ a d quad text{and} quad b c ]

Given that the x-intercepts are (h r, 0) and (h - r, 0), and the y-intercepts are (0, k √(r^2 - h^2)) and (0, k - √(r^2 - h^2)), for these to be equal, we need:

[ h r k sqrt{r^2 - h^2} quad text{and} quad h - r k - sqrt{r^2 - h^2} ]

Adding these equations, we get:

[ (h r) (h - r) (k sqrt{r^2 - h^2}) (k - sqrt{r^2 - h^2}) Rightarrow 2h 2k Rightarrow h k ]

Hence, for the x-intercepts to equal the y-intercepts, the center of the circle must be at the origin (h, k) (0, 0).

Conclusion

In conclusion, while x-intercepts and y-intercepts on a circle are not always equal, they are equal if the circle's center is at the origin. This unique scenario showcases the importance of the circle's center position in determining its intercepts with the coordinate axes. Understanding these concepts is crucial for anyone working with circles in the coordinate plane.

Keywords: circle intercepts, x-intercepts, y-intercepts