Math Homework...

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I have some math homework and have no idea how to do it D:

The focus and directrix of a parabola are given. Write an equation for each parabola.

1.) (2,5), y=1

2.) (-4,0), x=6

Work is needed. Thank you!

[-Griffin]

Alright, I've never heard of a 'directrix' or 'the focus of a parabola', or indeed done anything like this, but I'm bored so I'll do it anyway. -Looks up definition of Directrix and focus-

Alright, doesn't look too hard.

1)
Any parabola (which us English just call 'a quadratic graph') can be solved by knowing 3 points on it. Since any point on a parabola is equidistant from the focus and directrix, you can find as many points as you like. The most obvious one is vertically between them, so the point (2, 3) must lie on the line. There must also be two points with the same Y-value as the focus (the ones you'd hit if you just went along the X-axis, parallel to the directrix) which would be at (6, 5) and (-2, 5) (those are both 4 from the directrix and focus, in a straight line, since we already knew it has the same Y, 5, which was 4 away, the X is easy).

Substituting gives us:

3=4a+2b+c
5=36a+6b+c
5=4a-2b+c

The last two are equal, so:

36a+6b+c=4a-2b+c

Take c and simplify:

32a+8b=0

Divide by 8 and move the 'a' to the other side.

b = -4a

Substitute this into the first two equations to get:

3=c-4a
5=c+12a

Times the top one by 3 to get equal a values (prefer this to equal c here, since I'm doing it mentally)

9=3c-12a

Add them together:

14=4c

Divide by 4:

c=3.5

Sub that into "3=c-4a"

3=3.5-4a

Take 3.5, then divide by 4

a= [sup]1[/sup]/[sub]8[/sub]

Sub them both into "3=4a+2b+c"

3=2b+4

Take 4 from both sides and divide by 2.

b = -0.5

Final answer:

a= 0.125
b= -0.5
c= 3.5

Y=[sup]1[/sup]/[sub]8[/sub]X[sup]2[/sup] - [sup]1[/sup]/[sub]2[/sub]X + 3.5

I checked it with both other points and it works for them, so I'm reasonably confident here

EDIT: Something went seriously wrong, repeatedly, with my working on the second one :S You're on your own there.

[-Griffin]

Disregard the above edit.

I looked around, and 'a' in Y=aX^2+bX+c is always equal to 1/(2*distance from diretrex to focus.)

Subbing the point directly between them (1,0) gives you 1=0a+0b+c => c = 1

Because The focus has a Y coordiate of 0, B = 0.

a = 1/(2*10) = [sup]1[/sup]/[sub]20[/sub]

X = [sup]1[/sup]/[sub]20[/sub]Y[sup]2[/sup] + 1

= answer.

Checked with point (2, root 20) and it works fine.

The mistake was just really stupidly dropping a negative repeatedly x_X