How Do You Find Tangent and Normal Lines to a Parabola?

[bard]

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

2)The normal line to a curve c at a point Pis, by defininiton, the line that passes through p and is perpindicular to the tangent line c at P.

Where does the normal line to the parabola y=x^2-x at point (1,0) intersect the parabola a second time?

3) Find a parabola with equation y=ax^2+bx+c that has slope 4 at x=1, slope -8 at x=-1and passes through the point (2,15)

Thank You

 

[sol1]

Late Bloomer

http://archives.math.utk.edu/visual.calculus/

I have much to learn, and the converstaion is ahead of my knowledge but I am interested to follow as best I can. Sorry for intrusion.

Sol

 

[StephenPrivitera]

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

2)The normal line to a curve c at a point Pis, by defininiton, the line that passes through p and is perpindicular to the tangent line c at P.

Where does the normal line to the parabola y=x^2-x at point (1,0) intersect the parabola a second time?

3) Find a parabola with equation y=ax^2+bx+c that has slope 4 at x=1, slope -8 at x=-1and passes through the point (2,15)

Thank You

I'll break the rules and get you started with the first one. The equation for a tangent line:
y=f'(a)(x-a)+f(a)
You know,
-3=f'(a)(2-a)+f(a) holds for both lines
You also know,
f(a)=a^2+a
f'(a)=2a+1
So, -3=(2a+1)(2-a)+a²+a

 

[HallsofIvy]

Seriously, you should never ask for help until you have worked hard on the problem yourself. If you have done that, show us what you have done and where you think you got stuck. That way, our replies can be more specific and make more sense to you.

Stephen Privatera gave you a good start on the first problem (I won't chastise him TOO harshly).

For the second problem, you need to be able to write down the equation of the normal line. Of course, the derivative gives you the slope of the tangent line. How do you find the slope of the line perpendicular to the tangent line?

For problem 3, you need to find three numbers, a, b, and c and you have 3 conditions: the derivative at two points and the value at specific x. Find the derivative of ax^2+ bx+ c and plug in the values given for two equations, put the given values of x and y into the orginal equation to get a third equation. Solve those three equation for a, b, c.