A circle is one of the most fundamental shapes in geometry. It is defined as a set of all points that are equidistant from a common center point. Circles are encountered in various real-world applications, from physics to architecture.
In some problems, it is necessary to determine whether a circle touches both the x-axis and the y-axis. This can be helpful in analyzing the position and characteristics of a circle.
One straightforward approach to verify if a circle touches both axes is by examining its equation. The equation of a circle with center coordinates (h, k) and radius r is given by:
(x – h)^2 + (y – k)^2 = r^2
If a circle touches the x-axis, the y-coordinate of its center (k) will be equal to the radius (r). Similarly, if a circle touches the y-axis, the x-coordinate of its center (h) will be equal to the radius (r). By comparing the values of k and r as well as h and r, it can be determined if the circle touches both axes.
For example, consider a circle with equation (x – 3)^2 + (y + 4)^2 = 25. Here, the center of the circle is at point (3, -4) and its radius is 5. Since the radius (r) is equal to the absolute value of the center’s y-coordinate (4), the circle touches the x-axis. Additionally, the radius is equal to the absolute value of the center’s x-coordinate (3), indicating that the circle also touches the y-axis.
It is worth noting that a circle can touch just one axis or neither axis, depending on its position and characteristics. Understanding how to verify if a circle touches both the x-axis and the y-axis can be beneficial in solving various geometric problems and analyzing circles in different contexts.
Overview of the problem
When determining whether a circle touches both axes, we need to consider its equation in Cartesian coordinates. A circle in a 2-dimensional space is defined by the equation (x-a)^2 + (y-b)^2 = r^2, where (a, b) represents the coordinates of the center and r is the radius of the circle.
In order for the circle to touch both the x-axis and y-axis, we need to check if the coordinates of its center satisfy certain conditions. Let (a, b) be the coordinates of the center:
Conditions for touching the x-axis:
The circle touches the x-axis if and only if b = 0. This means that the y-coordinate of the center is zero.
Conditions for touching the y-axis:
The circle touches the y-axis if and only if a = 0. This means that the x-coordinate of the center is zero.
By applying these conditions, we can determine whether a circle touches both the x-axis and y-axis. If either condition is satisfied, the circle touches the corresponding axis. If neither condition is satisfied, the circle does not touch either axis.
Method 1: Distance formula
To verify if a circle touches both axes, you can use the distance formula. The distance formula calculates the distance between two points in a coordinate system.
Let’s assume the center of the circle is point (x, y). To check if the circle touches the x-axis, we need to find the distance between the center of the circle and a point on the x-axis. This point will have a y-coordinate of 0. Using the distance formula, the distance between two points (x1, y1) and (x2, y2) is:
d = √((x2 – x1)^2 + (y2 – y1)^2)
In our case, the distance between the center of the circle (x, y) and a point on the x-axis (x, 0) is:
d = √((x – x)^2 + (0 – y)^2)
Simplifying the equation, we get:
d = √(y^2)
Since the distance cannot be negative, if the value of y is equal to the radius of the circle, then the circle touches the x-axis.
To check if the circle touches the y-axis, we need to find the distance between the center of the circle and a point on the y-axis. This point will have an x-coordinate of 0. Using the distance formula, the distance between the center of the circle (x, y) and a point on the y-axis (0, y) is:
d = √((0 – x)^2 + (y – y)^2)
Simplifying the equation, we get:
d = √(x^2)
Again, if the value of x is equal to the radius of the circle, then the circle touches the y-axis.
By applying the distance formula, you can determine if a circle touches both axes. If the distances calculated in both cases are equal to the radius of the circle, then the circle touches both the x-axis and the y-axis.
Method 2: Equation of a circle
Another method to verify if a circle touches both axes is by using the equation of a circle. The equation of a circle with center (h, k) and radius r is given by:
Step 1:
- First, determine the equation of the circle using the given information.
Step 2:
- Substitute the values of the x-intercept and y-intercept into the equation of the circle.
- If the resulting equations are satisfied, then the circle touches both axes. Otherwise, it does not.
Let’s illustrate this method with an example. Suppose we have a circle with center (3, 4) and radius 5. The equation of this circle is given by:
(x – 3)2 + (y – 4)2 = 25
Now, let’s substitute the x-intercept (5, 0) and y-intercept (0, 9) into the equation of the circle:
(5 – 3)2 + (0 – 4)2 = 25 (Substituting x-intercept)
(0 – 3)2 + (9 – 4)2 = 25 (Substituting y-intercept)
By simplifying the equations, we get:
4 + 16 = 25 (x-intercept equation)
9 + 25 = 25 (y-intercept equation)
Since the resulting equations are not satisfied (4 + 16 ≠ 25 and 9 + 25 ≠ 25), we conclude that the circle does not touch both axes.
This method allows you to verify if a circle touches both axes by directly using its equation and substituting the coordinates of the x-intercept and y-intercept.
