tangent lines to parabola y = x^2 + x through (2, -3)

U

UMstudent

New member
Joined
Feb 22, 2007
Messages
4
Just need a little help on this problem I've been mostly teaching everything to myself out of the book because my instructor is horrible at teaching but it happens i can't find an example of this problem so thanks for any help I can get.

Find the equations of both lines through the point (2, -3) that are tangent to the parabola y = x^2+x

And i would greatly appreciate if I could get an explanation of the problem not just an answer so I know how to do this in the future. Thanks again.
 
Equation of the line is of the form

\(\displaystyle \L y+3 = m(x-2)\)

We want this line to be tangent to the parabola at some point (a,b). Then,

(i) \(\displaystyle b = a^2 + a\) (since the point (a,b) is on the parabola)

(ii) \(\displaystyle m = 2a +1\) (since the slope of the tangent line at (a,b) is the derivative of \(\displaystyle y=x^2+x\) evaluated at (a,b) )

(iii) \(\displaystyle b+3 = m (a-2)\) (since (a,b) is on the tangent line)

Substituting (i) and (ii) into (iii), you get

\(\displaystyle \L a^2 + a + 3 = (2a+1)(a-2)\)

There are two solutions for a, which give you two possible slopes, hence two tangent lines.
 
You must log in or register to reply here.
Top