How to verify is a circle touches the both axes

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.

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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.

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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:

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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:

  1. First, determine the equation of the circle using the given information.

Step 2:

  1. Substitute the values of the x-intercept and y-intercept into the equation of the circle.
  2. If the resulting equations are satisfied, then the circle touches both axes. Otherwise, it does not.
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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.

Mark Stevens
Mark Stevens

Mark Stevens is a passionate tool enthusiast, professional landscaper, and freelance writer with over 15 years of experience in gardening, woodworking, and home improvement. Mark discovered his love for tools at an early age, working alongside his father on DIY projects and gradually mastering the art of craftsmanship.

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