Finding Tangent & Normal Lines to a Parabola & Finding Parabola w/Given Conditions

• bard
Good luck.In summary, the conversation discusses finding equations for tangent and normal lines to a parabola, as well as finding a parabola with given conditions. Guidelines and hints are given for solving the problems.
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

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

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

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.

1. What is a parabola?

A parabola is a type of curve that is formed when a plane intersects with a cone. It is a symmetrical shape that resembles a U or a smile.

2. How do you find the tangent line to a parabola?

To find the tangent line to a parabola, you first need to find the derivative of the parabola's equation. Then, plug in the x-coordinate of the point where you want to find the tangent line. The resulting value will be the slope of the tangent line. Finally, use the point-slope form of a line to write the equation of the tangent line.

3. What is a normal line to a parabola?

A normal line is a line that is perpendicular to the tangent line at a given point on the parabola. It intersects the parabola at a 90-degree angle and shares the same x-coordinate as the point of tangency.

4. How do you find the normal line to a parabola?

To find the normal line to a parabola, you first need to find the tangent line at the desired point. Then, you can use the negative reciprocal of the slope of the tangent line to find the slope of the normal line. Finally, use the point-slope form of a line to write the equation of the normal line.

5. What conditions are needed to find a parabola's equation?

To find a parabola's equation, you need to know the coordinates of at least three points on the parabola. Alternatively, you can use the focus and directrix of the parabola or the vertex and the equation of the axis of symmetry to determine the equation.

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