Definition: A conic section is a curve formed by slicing a double cone with a plane; the possible shapes are circles, ellipses, parabolas, and hyperbolas.
What are the conic sections?
Slice a double cone at different angles and you get four families: the circle, ellipse, parabola, and hyperbola. Each has a standard equation, a center or vertex, and special points called foci.
How do you identify one?
- x2 and y2 with equal positive coefficients: circle.
- x2 and y2 with different positive coefficients: ellipse.
- x2 and y2 with opposite signs: hyperbola.
- Only one variable squared: parabola.
Worked example: x2 + y2 – 6x + 4y – 12 = 0. Complete the square: (x – 3)2 + (y + 2)2 = 12 + 9 + 4 = 25. It is a circle with center (3, -2) and radius 5.
What mistake is common?
Forgetting to add the same numbers to both sides when completing the square. Mathbot keeps the equation balanced step by step.
Example: x^2 + y^2 – 6x + 4y – 12 = 0 is a circle, center (3, -2), radius 5