# Intersection of line and curve

• B
• AngleWyrm
In summary: So the only solution is to read an approximated slope by inspection.In summary, the conversation is about finding the coordinates of an intersection between a tangent line and a curve, but without all the necessary information, the problem is not well-defined. The only possible solution is to approximate the slope of the tangent line by inspection.
AngleWyrm
TL;DR Summary
Trying to find coordinates of an intersection between a tangent line and a curve
I have a formula y=log(x)/log(0.9) which has this graph:

I want to find the intersection of this curve and a tangent line illustrated in this rough approximation:

The axes have very different scales, so the line isn't actually a slope of -1, it's just looks that way.

How can I figure out:
1). the actual slope of the line
2). the coordinates of the intersection

Why do don't you calculate dy/dx to get the slope of the tangential line ?

AngleWyrm
Yeah that works; looks like about -25/0.5 = -50 slope

AngleWyrm said:
How can I figure out:
1). the actual slope of the line
2). the coordinates of the intersection
If you know one of these you can determine the other, but if you don't know either then the problem is not well defined: there are an infinite number of tangent lines with some slope that intersect the curve somewhere.

AngleWyrm said:
Summary:: Trying to find coordinates of an intersection between a tangent line and a curve

I want to find the intersection of this curve and a tangent line illustrated in this rough approximation:
In the lower of your two graphs, it looks like you are concerned with the point (.2, f(.2)). Is this the point of intersection for this problem? If so, that is information you should have provided.

Mark44 said:
it looks like you are concerned with the point (.2, f(.2)). Is this the point of intersection for this problem?
No, that's an estimate of the answer to the question what are the coordinates.

AngleWyrm said:
No, that's an estimate of the answer to the question what are the coordinates.
Are you given the x-coordinate at the point of intersection? This is post #7, and we're still not sure exactly what you're asking.

If not, the problem is not well defined, as @pbuk said.
pbuk said:
If you know one of these you can determine the other, but if you don't know either then the problem is not well defined: there are an infinite number of tangent lines with some slope that intersect the curve somewhere.

AngleWyrm said:
Yeah that works; looks like about -25/0.5 = -50 slope
pbuk said:
If you know one of these you can determine the other
You know one of these. Carry on.

From post #5:
Mark44 said:
it looks like you are concerned with the point (.2, f(.2)).
AngleWyrm said:
No, that's an estimate of the answer to the question what are the coordinates.
Is x = .2 the known x-coordinate at the point of tangency? If so, the point (.2, f(.2)) is NOT an estimate.
AngleWyrm said:
You know one of these.
No we don't! You haven't said what it is that is known in this problem.
I'm about to give up here and lock this thread, unless you are more forthcoming with the information of this problem.

Last edited:
Do you want one of these lines ?

Mark44 said:
Is x = .2 the known x-coordinate

robphy said:
As others have said, there are an infinite number of tangent lines.
Are there an infinite number of tangent lines with a slope of -50?
Or does that restrict the answer set to one unique line and set of coordinates?

Last edited:
AngleWyrm said:
Re-read the thread summaryAre there an infinite number of tangent lines with a slope of -50?
Or does that restrict the answer set to one unique line and set of coordinates?
Say the condition is -50 slope, the solution of the equation
$$y'(x_0)=-50$$
gives unique tangential point ##(x_0,y_0)=(x_0,\frac{\log x_0}{\log 0.9})## and thus the equation of the line is
$$y-y_0=y'(x_0)(x-x_0)=-50(x-x_0)$$

AngleWyrm
Mark44 said:
Is x = .2 the known x-coordinate at the point of tangency?

AngleWyrm said:
I have read it several times. I've also asked you several times to tell us what information is given in this problem. Since you refuse to do so, I am closing this thread.

Looks like it's time to close this thread. While the problem is missing some key info or the OP is unwilling to provide it, the only possible solution is to read an approximated slope from the given graph by inspection.

Basically, to find the tangent line accurately for a known algebraic function, we need to know the function and the x point or y point of the tangent line of interest. While we have the equation for the curve, we don't know what x point or y point to use.

## 1. What is the difference between a line and a curve?

A line is a straight path that extends infinitely in both directions, while a curve is a smooth, continuous path that changes direction. Lines have a constant slope, while curves can have varying slopes.

## 2. How do you find the intersection point of a line and a curve?

To find the intersection point, you need to solve the equations of the line and the curve simultaneously. This can be done by setting the equations equal to each other and solving for the variables. The resulting point is the intersection point.

## 3. Can a line and a curve have more than one intersection point?

Yes, a line and a curve can have multiple intersection points. This occurs when the line crosses the curve at different points, or when the curve intersects itself.

## 4. What does the intersection point represent?

The intersection point represents the coordinates where the line and the curve meet. It is the point that satisfies both equations simultaneously.

## 5. How is the intersection of a line and a curve useful in real-life applications?

The intersection of a line and a curve has many practical applications, such as in engineering, physics, and economics. It can be used to determine optimal solutions, analyze data, and predict outcomes. For example, the intersection point of a demand and supply curve in economics represents the equilibrium price and quantity of a product.

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