Final answer:
To derive the equation of the parabola, calculate the distance between a general point on the parabola and the focus, then equate it to the distance between the point and the directrix. Square both sides of the equation and simplify to obtain the equation of the parabola in standard form. Rearrange the equation by completing the square and isolate y to obtain the final equation which is y = (-1/14)(x^2 + 10x + 25) + 63/14.
Step-by-step explanation:
The equation of a parabola with a focus at (-5, -5) and a directrix of y = 7 can be derived using the distance formula. The distance between a point (x, y) on the parabola and the focus is equal to the distance between the point and the directrix. Let's solve for the equation step-by-step:
- First, find the distance between a general point (x, y) and the focus (-5, -5) using the distance formula:
- d = sqrt((x + 5)^2 + (y + 5)^2)
- Next, find the distance between the general point (x, y) and the directrix y = 7:
- d = |y - 7|
- Since the distances are equal, we can equate the two expressions, resulting in an equation:
- sqrt((x + 5)^2 + (y + 5)^2) = |y - 7|
- Square both sides of the equation to eliminate the square root:
- (x + 5)^2 + (y + 5)^2 = (y - 7)^2
- Expand and simplify the equation:
- x^2 + 10x + 25 + y^2 + 10y + 25 = y^2 - 14y + 49
- Combine like terms to isolate x:
- x^2 + 10x = -14y - 14 - 49 + 25
- Further simplify the equation by multiplying through by -1:
- -x^2 - 10x = 14y + 14 + 49 - 25
- Divide both sides of the equation by -1 to obtain the standard form of the parabolic equation:
- x^2 + 10x = -14y + 38
- Rearrange the equation by completing the square:
- x^2 + 10x + 25 = -14y + 63
- Further rearrange the equation to isolate y:
- y = (-1/14)(x^2 + 10x + 25) + 63/14
Therefore, the equation of the parabola is y = (-1/14)(x^2 + 10x + 25) + 63/14.