# Is it possible for 2 tangent lines of this function to be perpendicular?

• unf0r5ak3n
In summary, it is possible to find two tangent lines to a function that are perpendicular to each other, but it is not possible for two lines to be tangent to a function and also be parallel to the function.
unf0r5ak3n

## Homework Statement

Is it possible that there are two different tangent lines to this function that are perpendicular to each other? If so, find the equations of the two lines and show that they are tangent to ƒ(x) and perpendicular to each other. If not, show why it is not possible.

ƒ(x) = x^2 + 2

## The Attempt at a Solution

I believe it is possible, I found the line tangent to ƒ(x) at x=2 and its perpendicular line was fairly close to a tangent to the function. However I'm not sure how to find the equations that would be exactly tangent to ƒ(x) as well as perpendicular to one another

Start writing a general equation for a tangent of f(x)

f(2) = 4x-2

A general equation f'(x) = 2x

right the derivative is 2x

Ok, so what can you say about the equations of 2 perpendicular lines ?
Equation like y= mx+q

they should be reciprocals

unf0r5ak3n said:
they should be reciprocals
The equations won't be reciprocals.

?? confused

unf0r5ak3n said:
they should be reciprocals

Be reciprocal is one change, the other change ?

You're given y= 3x+1

Write a perpendicular line.
y = ?

y = (x-1)/3

unf0r5ak3n said:
?? confused
The slopes of the two lines will be reciprocals.

so no 2 tangents can form a perpendicular line at f(x) = x^2+2?

unf0r5ak3n said:
so no 2 tangents can form a perpendicular line at f(x) = x^2+2?

Yes they can, and they do in an infinite number of points.
But you have to learn to write equations of perpendicular lines before.

what would be an example and how do you find the initial one?

unf0r5ak3n said:
what would be an example and how do you find the initial one?
An example of what? The initial one what? Please try to write sentences that are clear.

What would be an example of two equations of lines that are perpendicular and tangent to the function f(x)=x^2+2?
Would any line form a perpendicular angle as well as be tangent to the function? If not, how do you find the equation of the first line as appose to the second one?

TO THE OP: I think you may be misunderstanding the question you were asked. The question is: find two tangent lines to f(x) such that the two are perpendicular to each other. In other words, they will both be parallel to the function f(x) at their respective points, but they will be perpendicular to each other.

********************

Now, you already know that the derivative of f(x) = x^2+2 is f'(x) = 2x. THAT is the slope of the tangent line of f(x) at x.

The question can be rephrased as: Find two points x1 and x2 such that f'(x1) is the negative reciprocal of f'(x2).

Using the fact that f'(x) = 2x, this is equivalent to asking: Find two points x1 and x2 such that 2x1 is the negative reciprocal of 2x2. Can you find two such points?

like f(1)=2 and f(-.25)=-.5?

f'(x) for both*

Yes, exactly!

ok, but i don't understand what to use them for?

Well, f'(1) is the slope of the tangent line to f(x) at x = 1, and f'(-0.25) is the slope of the tangent line to f(x) at x = -0.25.

So, you are looking for two lines:

y = a1x + b1

and

y = a2x + b2

That are tangent to f(x) at x=x1 and x = x2, respectively, but which are perpendicular to each other.

You have just found the slopes that these tangent lines need to have! That is, you have found that a1 should equal 2, and a2 should equal -0.5. This automatically makes the two lines you are creating perpendicular to each other. Now you just need to choose b1 and b2 so that the first line intersects the point (x1, f(x1)) and the second line intersects the point (x2,f(x2)).

oh! i actually figured it out as you posted that last one comment, thank you so much!

curious if this is actually correct,
In the function f(x)=x^2+2, lines y=2x+1 and y=.5x+1.9375 are tangent to f(x) and are also perpendicular.

Last edited:
unf0r5ak3n said:
oh! i actually figured it out as you posted that last one comment, thank you so much!

curious if this is actually correct,
In the function f(x)=x^2+2, lines y=2x+1 and y=.5x+1.9375 are tangent to f(x) and are also perpendicular.
These lines aren't perpendicular. The slope of the first line is 2 and that of the second line is 1/2. These are reciprocals, but not the negative reciprocals of each other.

Also, the lines have to be tangent to the curve y = x^2 at points on this curve. I don't believe that the line y = 2x + 1 is actually tangent to this curve anywhere.

Your constants are wrong. Also, that .5x should be a -.5x.

You want to find a constant such that 2*x + b1 = x^2 + 2 for x = 2.
You also want to find a constant such that -0.5*x + b2 = x^2 + 2 for x = -0.25

## 1. Can two tangent lines of a function be perpendicular?

Yes, it is possible for two tangent lines of a function to be perpendicular. This can occur when the slope of one tangent line is the negative reciprocal of the slope of the other tangent line.

## 2. What conditions must be met for two tangent lines to be perpendicular?

For two tangent lines to be perpendicular, the slopes of the lines must be negative reciprocals of each other. This means that the product of the slopes must be -1.

## 3. Is there a visual representation of two perpendicular tangent lines?

Yes, a visual representation of two perpendicular tangent lines would be two lines intersecting at a right angle. This can be seen on a graph of a function where two tangent lines are drawn at a point where the tangent lines are perpendicular to each other.

## 4. Are there any real-life applications of perpendicular tangent lines?

Yes, perpendicular tangent lines have real-life applications in fields such as engineering and physics. For example, in a roller coaster design, the tracks must be tangent to the curves at a right angle in order for the ride to be safe and smooth.

## 5. Can perpendicular tangent lines occur at more than one point on a function?

Yes, it is possible for perpendicular tangent lines to occur at more than one point on a function. This can happen when the function has multiple points of inflection or when the function has multiple local maximum or minimum points.

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