Introduction to Circle Equations

Understanding the equation of a circle is fundamental in analytical geometry. This article will guide you through the process of finding the equation of a circle that touches the Y-axis and has a given center. We'll explore the standard form of a circle's equation and demonstrate its application to real-world scenarios.

Equation of a Circle

The standard equation of a circle is given by:

(x - h)2 (y - k)2 r2

Here, (h, k) represents the center of the circle, and r represents the radius.

Identifying Known Values

In this problem, we are given the following information:

The center of the circle is at (3, 4). The circle touches the Y-axis.

Knowing the circle touches the Y-axis means the distance from the center to the Y-axis is equal to the radius. The distance from the point (3, 4) to the Y-axis (the line x 0) is 3 units.

Calculating the Radius

The radius, r, is therefore 3 units. Substituting the values into the standard equation:

(x - 3)2 (y - 4)2 32

This simplifies to:

(x - 3)2 (y - 4)2 9

Interpreting the Equation Graphically

When a point on the Y-axis is tangent to the circle, it means the x-coordinate of this point is 0, and the y-coordinate is the same as the y-coordinate of the center, which is 4. The radius, r, remains 3 units.

Visualizing the Circle

On a Cartesian plane, the center of the circle is at (3, 4). The circle touches the Y-axis at the point (0, 4). This visualization helps to understand the relationship between the center and the tangency point.

Deriving the Equation

Since the circle is tangent to the Y-axis, we can derive the equation using the standard form:

(x - 3)2 (y - 4)2 32

This equation succinctly describes the circle with the given properties.

Final Equation

The equation of the circle, centered at (3, 4) and touching the Y-axis, is:

(x - 3)2 (y - 4)2 9

Further Exploration

For a deeper understanding, you can explore similar problems involving circles and other geometric shapes. Understanding the relationship between the center, radius, and tangency points will enhance your analytical skills in geometry.

Conclusion

By applying the standard form of a circle's equation, we were able to find the equation of a circle that touches the Y-axis and has its center at (3, 4). This process is a practical demonstration of how geometric knowledge can be used in various mathematical and real-world applications.