What formula should be used to derive the equation of a parabola given its focus and directrix?
• The distance formula ??
• The midpoint formula
• y = ax2 + bx + c
• The area formula for a circle
I'd say the distance formula, given the definition of a parabola.
To derive the equation of a parabola given its focus and directrix, you can use the distance formula. Here's how:
1. Start by understanding the definition of a parabola. A parabola is a curve where each point on the curve is equidistant from a fixed point called the focus, and a fixed line called the directrix.
2. Suppose the focus of the parabola is at the point F(a, b), and the directrix is a horizontal line y = c.
3. Take an arbitrary point on the parabola with coordinates P(x, y). The distance from P to the focus F is given by the distance formula:
√((x - a)² + (y - b)²)
4. The distance from P to the directrix is the perpendicular distance, which can be calculated as |y - c|.
5. Since P lies on the parabola, the distance from P to the focus will be equal to the distance from P to the directrix. Therefore, we can set up the equation:
√((x - a)² + (y - b)²) = |y - c|
6. Square both sides of the equation to eliminate the square root:
(x - a)² + (y - b)² = (y - c)²
7. Expand the equation:
x² - 2ax + a² + y² - 2by + b² = y² - 2cy + c²
8. Simplify the equation by canceling out the y² terms:
x² - 2ax + a² + b² = - 2cy + c²
9. Rearrange the equation to isolate the terms with x on one side:
x² - 2ax = - 2cy + c² - a² - b²
10. Finally, simplify the equation to obtain the standard form of the equation of a parabola:
x² + (4ac)x = 4ac² - a² - b² + c²
Therefore, the correct formula to derive the equation of a parabola given its focus and directrix is:
x² + (4ac)x = 4ac² - a² - b² + c².