# How Can I Find Tangents on a Surface Passing Through an External Point?

• AtulSharma
In summary, the problem at hand is to find the equation of some tangents on a 3D surface that pass through an external point. The approach suggested is to find the derivative of the function and use it to determine the slope of the tangent at any given point. From there, a relation between the coordinates of the point and the slope of the tangent can be established to find the desired equations. It is noted that this solution may not be applicable for complex surfaces with both concave and convex components.
AtulSharma
Finding Tangents... Interesting !

Hello,

I have an interesting problem for you.

I have a surface & an external point which is outside the surface.

I need to find the equation of all the tangents on this surface which pass through the external point.

Regards,
AtulSharma

AtulSharma said:
Hello,

I have an interesting problem for you.

I have a surface & an external point which is outside the surface.

I need to find the equation of all the tangents on this surface which pass through the external point.

Regards,
AtulSharma
Are you just trying to test us, or you really need an answer to this question?

I myself need an answer to the problem.

well, i am not sure whether my reasoning will be fine, because you have this surface, so if it were a curve it would most likely go like this.
You first find the derivative of that function, most likely you will have to differentiate it implicitly, so this way you will manage to find the slope of the tangent line at any point on that surface, curve. So now you have this external point call it $$(x_1,y_1)$$, so now the slope of the tangent will be

$$\frac{y-y_1}{x-x_1}=m$$, but now you also have the slope of the tangent line at any point dy/dx so

$$\frac{dy}{dx}=\frac{y-y_1}{x-x_1}$$, so you will manage to find a relation between $$x, and \ y$$ but you also have one relation given by the eq of the curve, so i think now you have to look for a solution, when this line only touches that curve. I am not sure whether my reasoning is okay, but wait for other replies as well!

Well, I think its a good way to try. Let's see other replies also.

Thanks for this solution.

Is this a 2D or 3D problem. If it's a 3D problem, are you looking for all the tangent planes that include the outside point, or only tangent lines that go through the point (in this case you end up with a cone)? If it's a 2D problem then you just end up with 2 lines. This is assuming the surface isn't complex, such as one that has both concave and covex components.

This is a 3D problem. I need to find some (not all) of the tangents passing through the external point.

## What is a tangent?

A tangent is a line that touches a curve at only one point, without crossing through it. It represents the instantaneous rate of change of the curve at that point.

## Why is finding tangents interesting?

Finding tangents is interesting because it allows us to understand the behavior of a curve at a specific point. It also has many practical applications in physics, engineering, and other fields.

## How do you find a tangent to a curve?

To find a tangent to a curve, you can use the derivative of the curve at the given point. The slope of the tangent line is equal to the slope of the curve at that point.

## What are some real-life examples of tangents?

Some real-life examples of tangents include the trajectory of a projectile, the rate of change of a growing population, and the slope of a hill at a certain point.

## What are the limitations of using tangents?

One limitation of using tangents is that they only represent the behavior of a curve at a single point and may not accurately reflect the overall behavior of the curve. Additionally, tangents cannot be found for some curves, such as discontinuous or non-differentiable curves.

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