Finding the eq. of all tangent lines on a curve

[Gus_Chiggins]

Homework Statement

Problem: Find the equations of all tangent lines to the curve
y = x + 2x so that also go through the point (3, 14).


2. Do not use a derivative



3. I don't even know where to start. I searched my book there isn't really available. Any help would be much appreciated

 

[danago]

Homework Statement

Problem: Find the equations of all tangent lines to the curve
y = x + 2x so that also go through the point (3, 14).


2. Do not use a derivative



3. I don't even know where to start. I searched my book there isn't really available. Any help would be much appreciated

Is that really supposed to be y=x+2x, or have you mis-typed something?

 

[Mark44]

What's the correct equation? The "curve" y = x + 2x is a straight line that doesn't go through (3, 14), so no tangent can go through this point either.

Should it be y = x^2 + 2x?

 

[Gus_Chiggins]

sorry everybody,

yes I meant to say

y=x^2 + 2x

sorry

 

[Mark44]

OK, now that we've gotten that out of the way...

Let $(x_0, y_0)$ be the point of tangency on the graph of the curve. BTW, you have drawn the graph, right?

At the point of tangency, the tangent line has to extend from $(x_0, y_0)$ to (3, 14).

Here is an outline of the steps you'll need to carry out for this problem:

1. Find the slope of the line from $(x_0, y_0) = (x_0, x_0^2 + 2x_0)$ to (3, 14).
2. By calculating the derivative and evaluating it at $x_0$, find the slope of the tangent line.
3. Equate the value you got in step 1 with the value from step 2, and solve for $x_0$. (I got two values for $x_0$.)
4. Find the associated y value for each value of $x_0$ from step 3.
5. Using each point $(x_0, y_0)$, find the equation of the line from $(x_0, y_0)$ to (3, 14). There are two distinct equations.

Is that enough of a hint?