# Equation of both lines that are tangent to the graph y=x^2

1. Feb 18, 2010

### francis21

1. The problem statement, all variables and given/known data
Question: Determine the equations of both lines that are tangent to the graph of f(x) = x2 and pass through point (1,-3).

2. Relevant equations
Some of the equations that I could use for this problem are:
y-y1=m(x-x1) (Point-slope Equation)

the derivative of the function f(x) from first principles (the limit of the difference quotient, as h approaches 0)

3. The attempt at a solution

First, I took the derivative of the function x2.

As a result,

f`(x) = 2x

But I'm not sure on how to go from here.

2. Feb 18, 2010

### danago

Lets say that the line is tangent to the curve at the point with coordinates (p,p2). Using the definition of the gradient of a line along with the fact that we know it passes through (1,-3) as well as (p,p2):

$$m = \frac{\Delta y}{\Delta x} = \frac{p^2+3}{p-1}$$

Since we know that the line is tangent to the curve with equation y=x2, whose derivative, as you noted, is y'=2x, it should be clear that the slope of the line can also be expressed as m=2p, since we originally defined the point (p,p2) as the point of tangency.

Can you go from there?

3. Feb 18, 2010