# How do you know if a line is tangent to a curve?

Apr 2, 2018

#### Explanation:

By solving the two equations you will get a point $\left(x , y\right)$ which lies on both the curve and the straight line.
if you got more than one point then this line will be intersecting and not a tangent to the curve.
then by finding the first derivative of the curve and substituting with the value of the point$\left(x , y\right)$
if it's value is equal to the slope of the straight line then this line is its tangent.

For example :
determine whether the line $y = 2 x - 1$ is a tangent to the curve $y = {x}^{2}$

1) Finding the intersection point :
by solving the two equation the intersection point will be $\left(1 , 1\right)$

2) Finding the first derivative of the curve function:
$y = {x}^{2}$
$y ' = 2 x$
By substituing with the value of $x = 1$
$y ' = 2$
Which is equal to the slope of the straight line $y = 2 x - 1$