Equation Of Circle With Centre At Origin

The Equation of a Circle with Centre at the Origin: A Comprehensive Guide

The equation of a circle, a fundamental concept in geometry and coordinate geometry, describes the set of all points equidistant from a fixed point called the center. Understanding this equation, particularly when the center is at the origin (0,0), is crucial for various mathematical applications. This comprehensive guide will delve into the equation of a circle with its center at the origin, exploring its derivation, properties, and applications. We'll also examine how to manipulate this equation to solve various geometric problems.

Understanding the Distance Formula

Before diving into the equation of a circle, let's revisit the distance formula. This formula is the cornerstone of understanding the circle's equation. The distance between two points (x₁, y₁) and (x₂, y₂) in a Cartesian coordinate system is given by:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

This formula is derived directly from the Pythagorean theorem. Consider a right-angled triangle formed by the two points and their projections onto the x and y axes. The distance between the points is the hypotenuse of this triangle.

Deriving the Equation of a Circle at the Origin

Now, let's consider a circle with its center at the origin (0,0) and a radius 'r'. Any point (x, y) on the circumference of this circle is equidistant from the center, and this distance is equal to the radius 'r'. Applying the distance formula between the center (0,0) and a point (x, y) on the circle, we get:

r = √[(x - 0)² + (y - 0)²]

Simplifying this equation, we obtain the standard equation of a circle with its center at the origin:

x² + y² = r²

This simple yet powerful equation encapsulates all the points (x, y) that lie on the circle. The radius 'r' is a constant that determines the size of the circle.

Properties of the Equation x² + y² = r²

The equation x² + y² = r² possesses several key properties:

• Symmetry: The equation is symmetric about both the x-axis and the y-axis. If (x, y) is a point on the circle, then so are (-x, y), (x, -y), and (-x, -y).

• Circular Shape: The equation defines a perfect circle centered at the origin with a radius 'r'.

• Radius: The square root of the constant term on the right-hand side (r²) represents the radius of the circle.

• Center: The center of the circle is always located at the origin (0, 0).

• Intercept: The x-intercepts are found by setting y = 0, which gives x = ±r. Similarly, the y-intercepts are found by setting x = 0, which gives y = ±r.

Applications of the Equation of a Circle at the Origin

The equation x² + y² = r² finds widespread applications in various fields:

1. Geometry Problems:

The equation simplifies the solution of many geometric problems involving circles. For instance, determining if a point lies inside, outside, or on the circle becomes a simple matter of comparing the calculated distance from the origin to the radius.

• Point-Circle Relationship: If x² + y² < r², the point (x, y) lies inside the circle. If x² + y² > r², the point lies outside the circle. If x² + y² = r², the point lies on the circle.

• Finding Intersections: This equation can be used in conjunction with other equations (lines, other circles, etc.) to find points of intersection.

2. Trigonometry:

The equation is intrinsically linked to trigonometric functions. Consider a point (x, y) on the circle. Then, x = r cos θ and y = r sin θ, where θ is the angle between the positive x-axis and the line connecting the origin to the point (x, y). Substituting these into the equation x² + y² = r², we get:

(r cos θ)² + (r sin θ)² = r²

This simplifies to:

r²(cos²θ + sin²θ) = r²

Since cos²θ + sin²θ = 1 (a fundamental trigonometric identity), the equation holds true.

3. Physics and Engineering:

The equation finds use in many physics and engineering applications:

• Circular Motion: Describing the path of an object undergoing uniform circular motion.

• Wave Propagation: Modeling circular wavefronts emanating from a point source.

• Optics: Analyzing the geometry of lenses and mirrors.

4. Computer Graphics:

In computer graphics, the equation is essential for:

• Drawing Circles: Algorithms used to draw circles on a screen utilize this equation or its variations.

• Collision Detection: Determining whether two circular objects have collided.

Transformations of the Circle Equation

While the equation x² + y² = r² represents a circle centered at the origin, understanding transformations allows us to describe circles with centers elsewhere.

A circle with center (h, k) and radius 'r' has the equation:

(x - h)² + (y - k)² = r²

This is a translation of the circle at the origin, shifting it 'h' units horizontally and 'k' units vertically.

Solving Problems Involving the Equation of a Circle at the Origin

Let's illustrate the application of the equation through some examples:

Example 1: Determine if the point (3, 4) lies inside, outside, or on the circle x² + y² = 25.

• Substituting x = 3 and y = 4 into the equation: 3² + 4² = 9 + 16 = 25. Since this equals the right-hand side, the point (3, 4) lies on the circle.

Example 2: Find the radius of the circle given by the equation x² + y² = 49.

• Comparing this equation to x² + y² = r², we can see that r² = 49. Therefore, the radius r = √49 = 7.

Example 3: Find the x-intercepts of the circle x² + y² = 16.

• To find the x-intercepts, we set y = 0: x² + 0² = 16, which gives x² = 16, so x = ±4. The x-intercepts are (-4, 0) and (4, 0).

Example 4: A circle passes through the point (5, 12). What is the equation of the circle if its center is at the origin?

• The distance between the point (5, 12) and the origin (0, 0) is the radius. Using the distance formula: r = √[(5-0)² + (12-0)²] = √(25 + 144) = √169 = 13. Therefore, the equation of the circle is x² + y² = 169.

Conclusion

The equation of a circle with its center at the origin, x² + y² = r², is a fundamental concept with broad applications across various disciplines. Understanding its derivation, properties, and transformations is crucial for tackling numerous geometric problems and solving practical applications in diverse fields, ranging from geometry and trigonometry to physics, engineering, and computer graphics. By mastering this seemingly simple equation, you unlock a powerful tool for solving a wide range of mathematical and real-world problems. Remember to practice solving various problems to solidify your understanding and build your problem-solving skills.