how is y = −1/2 (x−2)² + 5/2 when the parabola given directrix y=3 and focus F=(2,2)?
i dont understand this, i kept using the distance formula but other people are using formulas that i dont know.people keep giving me that as the equation but i get a different answer that is wrong. can you please explain why that equation is the answer? thanks
More Questions From This User See All
Answers & Comments
Verified answer
So, the parabola is the locus of points whose distance from (2,2) is the same as their distance from the line y=3. If such a point is denoted by (x,y), then we know
(x-2)^2+(y-2)^2 = (y-3)^2
The above equation has, on the left, the square of the distance from (x,y) to (2,2), and on the right, the square of the distance from (x,y) to the line y=3. Now we expand and simplify:
(x-2)^2 + y^2 - 4y + 4 = y^2 - 6y + 9
(x-2)^2 = 5 - 2y
2y = 5 - (x-2)^2
y = (5/2) - (1/2) (x-2)^2 [the same as the answer you are asking about]