Can you determine the circular objects in your surroundings? Furthermore, if these circular objects are placed on a table, can you describe the location of that circular object? This is where the equation of a circle comes into play. The equation of a circle does not represent the area of a circle equation. Instead, it provides an algebraic way to describe the circle’s position or the family of circles in a Cartesian plane. It contains only the coordinates of the center, a fixed point inside the circle, and the radius, which is the distance from the center to the boundary of the circle. Moreover, the circle equation denotes all the points lying on its circumference. So what is the equation of a circle? Equation of a circleAfter learning the equation of a circle, let us learn how to derive the standard equation of a circle. The circle’s center coordinates are denoted by (a, b) as shown in the figure, and the radius is represented by r and (h, k) are the arbitrary points located on the circle’s circumference. (h – a)²+ (k – b)² = r², which is the standard equation of the circle. The general notation for representing the circle uses x and y as arbitrary points. Therefore, the equation of the circle becomes (x – a)² + (x – y)² = r². Various forms of representing the equation of a circleDo you know the circle equation can be represented in multiple ways? The equation of a circle can be expressed in various forms depending on the circle’s position in the Cartesian plane. The multiple forms of representing the circle are: General form Standard form (x – x1)² + (y – y1)²= r², where (x, y) is the arbitrary coordinates on the circumference of the circle, r is the radius of the circle, and (x1, y1) are the coordinates of the center of the circle. The standard form of the equation of the circle is derived from the distance formula. Parametric form Polar form Polar form representation is similar to the parametric form of the circle equation. The polar form is mostly used to represent the equation of the circle whose center is at the origin. For this, take an arbitrary point A having coordinates (r cos𝜃, r sin𝜃) on the periphery of the circle and a radius r, which is the distance between the random point and the origin. The equation of the circle having radius A and center at the origin will be given by, x²+ y² = A². Putting the values of x = r cos𝜃 and y = r sin𝜃, we get, (r cos𝜃)² + (r sin𝜃)² = A² r²cos²θ + r²sin²θ = A² r²(cos²θ + sin²θ) = A² r²(1) = A²(Since, cos²θ + sin²θ = 1 from the trigonometric identities) r = A where p is the radius of the circle. Thus, the polar form is used to find the radius of the circle from the standard form of the equation of the circle. Steps to find the equation of a circleWe have seen different ways of representing the equation of a circle depending upon the position of the center of the circle in the Cartesian plane. Therefore, to write the equation of a circle, when the coordinates of the center are given, one can follow these steps: Step 1: Figure out the coordinates of the center of the circle (x1, y1) and the radius of the circle. Understanding the equation of a circle with examplesExample 1: Find the equation of a circle whose center passes through the origin. Example 2: What will be the general equation of a circle whose radius is 6 units and the center lies on (4, 2)? x² + y² – 8x – 4y = 16. Example 3: Find the radius of the standard equation by converting the following standard equation of a circle into the polar form: x² + y²= 25. Therefore, the radius of the circle is 5 units. Example 4: What is the value of the center of the circle and radius if the standard equation of the circle is (x + 7)² + (y – 9)² = 529? x1 = -7, y1 = 9 and r = 23. Rewrite in standard form and graph.
Given a circle in general form, determine the intercepts. Given the graph of a circle, determine its equation in general form. |