Find equation of a tangent y=x^2 through a point (a,b) where (a,b) is not on a graph

This is a project in my calc 1 class. we are allowed to get help from others, but i couldn't find a tutor in our math lab that could help.

What I know: I know how to find the equation of tan line of a point thats on the graph. i would take derivative of y=x^2, which is dy/dx=2x. if i took a point lets say (2,2) on graph of y=x^2, i would take the x=2 and imput into derivative, which would give me the slope , then imput my values into y-y1=m(x-x1) simplify and get the equation of a tan line. What im confused is how to do with a point that is not on graph of y=x^2. Am i on the right track, or am i totally confused about what i need to do. Any suggestions and or tips would be greatly appreciated. Thanks ahead of time, Matt

Re: Find equation of a tangent y=x^2 through a point (a,b) where (a,b) is not on a g

You do exactly as you describe, but instead of (2,4) or (3,9), use $\displaystyle (x_0,x_0^2)$. So you have one equation representing a whole bunch of lines - it depends on what you choose for $\displaystyle x_0$.

Now you plug in your point and solve for $\displaystyle x_0$. That gives you the equation for the line and the point where it's tangent to the parabola.

Not all points will work, by the way. If you try to plug in, say, (1,2), you should find that there's no $\displaystyle x_0$ that works.

- Hollywood

Re: Find equation of a tangent y=x^2 through a point (a,b) where (a,b) is not on a g

Thanks for the help . I was able to finish this part . The next step says let P be a point (x,y) on parabola and let (a,b) be a point not on parabola . A) rewrite p in terms of x only . B) find slope of tangent to parabola at point p . C) calculate the slope between p and (a,b) . I'm not really sure what part A is asking . I think can do the other steps if I wasable to do part A.

Re: Find equation of a tangent y=x^2 through a point (a,b) where (a,b) is not on a g

Replace y with the definition of the parabola. This sounds like the same thing as the first post, and I am somewhat confused in that if your were able to complete that, why this has you stymied.

Re: Find equation of a tangent y=x^2 through a point (a,b) where (a,b) is not on a g

I guess I'm not sure what it is asking . My biggest struggle with math is the terminology and vocabulary . Thanks for the help . Matt

Re: Find equation of a tangent y=x^2 through a point (a,b) where (a,b) is not on a g

Part (A) is really simple - that might be what's confusing you. They want you to observe that a point p=(x,y) on your graph is of the form $\displaystyle (x,x^2)$. I used $\displaystyle x_0$ instead of $\displaystyle x$ because I wanted write the equation for the line in terms of x and y.

So can you do parts (B) and (C) now?

- Hollywood