Finding tangent lines of a parabola

Question

Show that there are two tangent lines to the parabola y=x² that pass through the point P(0,-4). Find the coordinates of the points were this tangent lines intersect the parabola.

 

Source: CALCULUS |Early transcendentals| 6^th Edition  _{BY: JAMES STEWART}  page182 exercise 59.

Answer 1

This might be kind of hard to describe without being able to clearly format equations, but here goes. 

 

First, sketch it.  You'll see that there are two lines which will come up from (0,-4) and touch the parabola.  If we can find enough properties of these lines, we can find out what the points of tangency are. 

 

Second, we know that the slope of the lines is constrained by the derivative, which is y' = 2x, which is the slope.

 

Third, we get the equation for slope from way back.  Call the point of tangency (x,y).  I'd normally use subscripts but I can't do that here, but in this case it won't hurt too much to use x and y for both general variables and the specific values we want. 

 

Using the given point, we can get an equation for slope, with one side being the one from algebra and the other being the derivative:

 

          (y-(-4))/(x-0) = 2x, which simplifies to

          (y+4)/x = 2x.

 

But we also know we are on the parabola, so we have y = x^2

 

Now you have two equations and two unknowns and can solve.  You'll get two x values which correspond to each of the two tangent lines.

 

After you get the x values, calculate the slope based on the derivative and make sure the lines will really go through (0,-4).  That's your check to see if you did it right.