Question

What is the equation, in standard form, of a parabola that contains the following points (-2,-20), (0,-4), (4,-20)?

Answer 1

See below.

Explanation:

A parabola is a conic and have a structure like

`f(x,y)=a x^2+ b x y +c y^2+d`

If this conic obeys the given points, then

`f(-2,-20) =4 a + 40 b + 400 c + d =0`
`f(0,-4)=16 c + d = 0`
`f(4,-20)=16 a - 80 b + 400 c + d = 0`

Solving for `a,b,c` we obtain

`a = 3d, b=3/10d,c=d/16`

Now, fixing a compatible value for `d` we obtain a feasible parabola

Ex. for `d=1` we get `a=3,b=3/10,c=-1/16` or

`f(x,y) = 1 + 3 x^2 + (3 x y)/10 - y^2/16`

but this conic is a hyperbola!

So the sought parabola has a particular structure as for instance

`y=a x^2+bx+c`

Substituting for the previous values we get the conditions

`{(20 + 4 a - 2 b + c = 0 ),(4 + c = 0),(20 + 16 a + 4 b + c =0):}`

Solving we get

`a=-2,b=4,c=-4`

then a possible parabola is

`y-2x^2+4x-4=0`