Finding the slope of line tangent to a parabola

In summary, To find the equations of lines through the point (2,-3) that are tangent to the parabola y=(x^2)+x, take the derivative of the function and set it equal to the slope of the line passing through (2,-3). This will give you a quadratic equation which will have two solutions. These solutions will be the x values at which the lines intersect the parabola. One line will be tangent to the parabola at (2,-3) and the other will be normal to the parabola at (2,-3).
  • #1
oates151
11
0

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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  • #2
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:
  • #3
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) .
 
  • #4
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)
 
  • #5
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
 

1. How do you find the slope of a line tangent to a parabola?

To find the slope of a line tangent to a parabola, you must first differentiate the equation of the parabola. This will give you the derivative function, which represents the slope of the tangent line at any given point on the parabola. Then, you can plug in the x-value of the point of interest into the derivative function to find the slope of the tangent line.

2. What is the significance of the slope of a line tangent to a parabola?

The slope of a line tangent to a parabola represents the rate of change of the parabola at a specific point. It tells you how fast the parabola is increasing or decreasing at that point, and can also be used to find the equation of the tangent line.

3. Can there be more than one tangent line to a parabola at a given point?

No, there can only be one tangent line to a parabola at a given point. This is because the tangent line must touch the parabola at only one point and have the same slope as the parabola at that point. Therefore, there can only be one unique tangent line for each point on a parabola.

4. How does the slope of a line tangent to a parabola change as you move along the curve?

The slope of a line tangent to a parabola changes as you move along the curve because the rate of change of the parabola is constantly changing. At the vertex of the parabola, the slope of the tangent line is 0, and as you move further away from the vertex, the slope either increases or decreases depending on the direction of the parabola's curvature.

5. Can the slope of a line tangent to a parabola ever be undefined?

Yes, the slope of a line tangent to a parabola can be undefined at the vertex of the parabola. This is because the derivative function, which represents the slope of the tangent line, has a vertical tangent at the vertex, meaning it has no defined slope at that point.

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