Suppose that on a parabola with vertex V and focus F there exist a point A such that AF = 20 and AV = 21. What is the sum of all possible values of the length FV?

The sum of the two possible values of the length FV is 40/3.

Expert Answers

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Denote the distance `FV ` as `p gt0 . ` We can assume that the vertex is at the origin and the focus is at `( 0 , p ) . ` Then the parabola has the equation `y = x ^ 2 / ( 4 p ) ` or `x ^ 2 = 4 p y , ` and its directrix is the straight line with the equation `y = -p .`

By the definition of parabola we know `AF = y + p = 20 ` (the distance from any point to the vertex is the same as the distance to the directrix). Also it is given that `x ^ 2 + y ^ 2 = 21 ^ 2 , ` and we already know that `x ^ 2 = 4 p y . ` This gives `4 p y + y ^ 2 = 21 ^ 2 ` and `y = 20 - p , ` so

`p ^ 2 - 40 p + 20 ^ 2 + 80 p - 4 p ^ 2 = 21 ^ 2 ` or `3 p ^ 2 - 40 p + 41 = 0 .`

It is the same as `p ^ 2 - 40 / 3 p + 41 / 3 = 0 , ` where Vieta's formula tells us that `p_1 + p_2 = 40 / 3 .`

Last Updated by eNotes Editorial on February 26, 2021
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