PRE CALCULUS - CIRCLE - CONIC SECTIONS | Standard Form to General Form of the Equation of the Circle

The video introduces the topic of converting a circle's equation from standard form (x-h)² + (y-k)² = r² to general form x² + y² + Dx + Ey + F = 0. This is the third video in a series on circles under conic sections.

The first example demonstrates converting (x+2)² + (y-3)² = 64. The process involves expanding the squared binomials. For (x+2)², the shortcut a² + 2ab + b² is used, resulting in x² + 4x + 4. manual expansion using FOIL method is also shown for clarity.

The expression (y-3)² is expanded to y² - 6y + 9. The expanded terms are then combined with the constant on the right side of the equation (64). The terms are regrouped, and 64 is transposed to the left side, becoming -64. The constants (4 + 9 - 64) are simplified to -51.

The terms are arranged in the general form: x² + y² + 4x - 6y - 51 = 0. This concludes the first example, demonstrating the conversion to the general form.

The second example involves converting (x+1)² + (y-2)² = 49. Viewers are encouraged to pause and try solving it themselves. Expanding (x+1)² results in x² + 2x + 1. Expanding (y-2)² results in y² - 4y + 4. The terms are then combined and rearranged.

The combined terms are arranged to x² + y² + 2x - 4y. The constants (1 + 4 - 49) are simplified to -44. The final general form is x² + y² + 2x - 4y - 44 = 0. The video concludes by encouraging viewers to watch a related video on converting from general to standard form and to subscribe to the channel.