How do you show that a tangent line does not pass through a certain point? The original function is f(x) = x2. I know that the equation for the slope of the tangent line is f(x) = 2x. I do not know however, how to show that this tangent line does not pass through the point, (-2, 5).

Question

phi · Answer

You can do this:    
write the equation of the tangent line in point-slope form:
$$ y - y_0= m(x - x_0) $$
as you state, the slope m is a function of x, i.e.  m= 2x.
also, (-2,5) is a point on the line.  Using this info we have
$$ y - 5 = 2x(x+2) \ y = 2x^2 +4x + 5 $$
If this "line" intersects the curve y= x^2, we can find the intersection point (x,y) by setting y values equal and solving for x:
$$  2x^2 +4x + 5 = x^2 \ x^2 +4x+5=0 $$
using the quadratic formula, the roots are
$$ x= -2 \pm i $$
In other words there is no real x that satisfies the equation, and therefore no line that is both tangent to $ y=x^2 $ and contains the point (-2,5).