Equation of a circle
Introduction to the Equation of a Circle
- A circle in a coordinate plane is defined by its centre and radius.
- The equation of a circle with centre at the origin (0,0) is x² + y² = r², where r is the radius of the circle.
- If the circle is not at the origin, and its centre is at point (h,k), the equation changes to (x-h)² + (y-k)² = r².
The Centre and Radius of a Circle
- The values of (h,k) give the coordinates of the centre of the circle in the equation.
- The radius of the circle is the square root of the constant term (r²) in the equation, which is positive.
Solving Problems Involving the Equation of a Circle
- The given equation of a circle can often be rearranged to make it clear in the standard form.
- It may be necessary to complete the square to rewrite the equation in the standard form.
Example Problem
- Consider the equation x² + y² - 6x + 8y - 9 = 0.
- To identify the centre and the radius, rewrite the equation by regrouping terms and completing the square to get the form (x-h)² + (y-k)² = r².
- This simplification leads to (x - 3)² + (y + 4)² = 4.
- In this case, the centre of the circle is at (3, -4) and the radius is 2 (since √4 = 2).
Final Notes
- It’s important to check your answers by substituting them back into the initial equation.
- In question scenarios, ensure to properly interpret the data given to accurately find the centre and radius of the circle.
- Proficiency in completing the square is crucial for working with equations of circles.