Question

If c > (1/2), how many lines through the point (0,c) are normal lines to the parabola y = x^2? What if c < or = (1/2)? Please explain the steps...

Answer 1

is this a calculus question?

Answer 2

if c=1/2 then there is only one line... that is the y-axis. if c<1/2, there are no lines because the parabola will always be positive. Now if c>1/2, there are always two lines.

Answer 3

how do you figure

Answer 4

take the derivative of y=x^2 to get y'=2x. This gives the slope of the tangent line for what ever x coordinate you look at that's on the parabola. Since you want the line that's normal to the parabola, take the negative reciprocal of the derivative to get you the slope of the line perpendicular to the tangent line. Since you know the line goes through (0, c) you can use that as the point to get the equation of the line perpendicular to tangent line using point-slope form of a line.

you should end up with this equation:

y = c - 1/2. this is how I came to the conclusions in my previous post.

Answer 5

so the slope of the normal line is -1/(2x)??

Answer 6

yep

Answer 7

i thought the slope had to be a number

Answer 8

like so that's equal to a slope of -1/2?

Answer 9

yes but the slope on a parabola depends on what point you look at. so the slope is given as an expression.

Answer 10

ohh for real ok

Answer 11

did you get to this point of the problem: y = c- 1/2?

Answer 12

i don't know i mean y-c = -(1/2x)(x-0)? or -(1/2)(x-0) ???

Answer 13

...the second right

Answer 14

yes... you got it now just simplify the right side to -1/2*x

Answer 15

right obviously so ok then yeah y = c-1/2*x

Answer 16

sorry, i mean -1/2

Answer 17

no x's

Answer 18

wait what

Answer 19

simplifying the right side, the x's cancell out right? you're left with only -1/2

Answer 20

where'd the x go

Answer 21

oh ok so it IS -1/2*x(x-0) right to begin with

Answer 22

just to be more clear, the right side is (-1/(2x))*(x-0)

Answer 23

right right that's what i meant ok

Answer 24

thanks

Answer 25

np

Answer 26

so if c = 1/2, y = 0, and... how does that mean that there is one line and that it's the y-axis?

Answer 27

when y=0, this makes x=0. so the line normal to the parabola at (0,0) is the vertical line, x=0, and this is the only one when c=1/2

Answer 28

oh.. k. so if c < 1/2, it's negative, and x^2 is always positive on y... how would i check that otherwise though if i didn't know the graph of the function?

Answer 29

from your equation, y=x^2, notice that x^2 is always positive... so y must always be positive.

Answer 30

but uh i'm sorry i know i must seem stupid as hell but how does that (c = 1/2, y = 0, x = 0, etc) mean that x is always 0 on the line? that's not just a point? geez....

Answer 31

that i know i was asking like if i didn't know the graph of the function like i know the graph of x^2... but obviously it's no big deal, same deal, it's gotta fit in the domain/range..... but uh there i'm a little fuzzy anyway what the hell is c in the normal line

Answer 32

it does not mean that x is always zero, all I did was to solve y=x^2 when y=0.
Now, if c>1/2, notice that your y is positive so that yields two x values, one positive and the other negative.

Answer 33

alright but so what is that you're finding? how is the resulting x,y a LINE and not a POINT? you know what i mean? that's how i'm confused

Answer 34

oh they're each other points on the normal line that goes through the point (0,c) so that makes a line

Answer 35

one for each result i got it sorry

Answer 36

remember the question was "how many lines..." ? the resulting points gave you the other point(s) needed to connect with (0, c) and be normal to the parabola...

Answer 37

gotcha

Answer 38

nice problem... thank you

Answer 39

aw no thank you