Finding the slope of line tangent to a parabola

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Homework Help Overview

The problem involves finding the equations of lines through a specific point that are tangent to a given parabola, y = x² + x. The original poster attempts to determine the slopes of these lines and has encountered confusion regarding the points of tangency.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the method of using derivatives to find slopes and question the interpretation of the problem regarding the point of tangency. There is a suggestion to set up a quadratic equation based on slopes derived from the tangent and the given point.

Discussion Status

The discussion reveals a lack of clarity about the points involved, particularly whether the lines are tangent to the parabola at the specified point or another point. Some participants have pointed out that the parabola does not pass through the original point given by the poster, prompting further exploration of the problem's setup.

Contextual Notes

There is confusion regarding the point (2, -3) and its relation to the parabola, as it has been noted that the parabola does not intersect this point, which may affect the problem's assumptions.

oates151
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Homework Statement



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

Homework Equations





The Attempt at a Solution



Took the derivative and got a slope of 5 and the slope of the normal line being -1/5, but the answer was marked wrong. How do I do this?

Two equations I got
y=-(1/5)x-(13/5)
y=5x-13
 
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Choose a point (a,y(a)). A line that goes through this point, AND the given point must have a slope of [itex]\frac{y(a) - (-3)}{a - 2}[/itex] Also, the slope at point 'a' can be given by the derivative of the function. This gives you two equal expressions for the slope in terms of a. It will be a quadratic equation. The roots will be the x values at which the lines intersect the parabola.
 
Last edited:
What do you mean? "both lines"

One of your lines is tangent to the parabola at (2, -3) .

The other is normal to the parabola at (2, -3) .
 
SammyS said:
What do you mean? "both lines"

One of your lines is tangent to the parabola at (2, -3) .

The other is normal to the parabola at (2, -3) .

Actually I think it's tangent to the parabola at (2, 6)
 
ElijahRockers said:
Actually I think it's tangent to the parabola at (2, 6)
Ha!

Yup, the parabola doesn't pass through (2, -3) ! DUH
 

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