Question

There are two tangent lines to the parabola y=x^2 that pass through the point (0,-4). Find the coordinates of the points where these tangent lines intersect the parabola.

Answer 1

I know f'(x) is 2x... and i know point slope formula... but I have no idea what to do... @Kainui

Answer 2

Except just trying points and finding the formula for a tangent line which has a y intercept of -4, but that seems inefficient and not what I'm supposed to be doing

Answer 3

How about this.
Set the equation of the tangent line as y=mx-4

Answer 4

Ok

Answer 5

Now, solve the system of equations.

y=mx+b
y=x^2

Answer 6

You would get\[x^2=mx-4 \rightarrow x^2-mx-4=0\]

Answer 7

Wouldn't it be x^2-mx+4?

Answer 8

Oops
Yeah. My bad.

Answer 9

\[x^2-mx+4=0\]

Answer 10

Now, because y=mx-4 and y=x^2 are tangent of each other, how many intersections do they have?

Answer 11

A single point
But two, one on each side of the parabola

Answer 12

For a quadratic equation, if it has one solution, its discriminant has to be 0, right?

Answer 13

Right

Answer 14

What is the discriminant for x^2-mx+4=0

Answer 15

Actually, can't you plug in 2x there for the m in x^2-mx+4, which will give you the x coordinates of the tangent points? Then plug those in to get the y's?

Answer 16

You can't do that.
The reason is 2x represents the slope of a tangent line of an ARBITRARY point on y=x^2
Because there are only 2 points where these tangent lines intersect the parabola, you can't do that.

Answer 17

Right, but that arbitrary point is the same arbitrary point that's being solved for with the x in the substituted formula from the system which is solving for that point

Answer 18

I tried it and got the right answer, +-2,4

Answer 19

m should be 4, -4

Answer 20

I worded it wrong.
It's not arbitrary.
2x means for ANY point on y=x^2

Answer 21

x^2-mx-4=0 doesn't hold true for ANY points on y=x^2

Answer 22

But that m is there because it was originally part of the equation for the slope of the tangent line, which by definition, has the same slope as the tangent point on the parabola

Answer 23

m was meant to be the slope of the tangent lines THAT PASSES THROUGH (0,4)

Answer 24

The line that passes through 0,4 that we are forcing to have the same slope as the parabola at some point x in order to be tangent, therefore has a slope of 2x as well

Answer 25

I agree. But, you have to understand that x is a variable, meaning that it changes.

Answer 26

The tangent point doesn't change, meaning that it has to stay at one place.

Answer 27

But it still represents the same point
We just don't know what that point is

Answer 28

And therefore are using x to represent it

Answer 29

We already used 'x' in y=x^2
You can't use 'x' again.

Answer 30

'x' is the variable that is used in functions.

Answer 31

The correct way of doing this your way would be this:

Answer 32

I intentionally represent only one tangent line for my own conveniance.