The conmen tangent between two circles

[shandymilo]

How do I find using differentiation the conmen tangent between two circles
x² +y²= 16 and (x-6)²+y²​= 4

 

[Ishuda]

How do I find using differentiation the conmen tangent between two circles
x² +y²= 16 and (x-6)²+y²​= 4

Differentiate the first to get
y₀' = -x₀/y₀
where (x₀, y₀) is a point on the first circle.

Differentiate the second to get
y₁' = -(x₁ - 6) / y₁
where (x₁, y₁) is a point on the second circle.

Now, if the line connecting the two points is a common tangent of the two circles what does that say about the relationship between (x₀, y₀) and (x₁, y₁)?

 

[shandymilo]

That they are equal one another?

 

[Ishuda]

That they are equal one another?

If there is a common line then the slopes are the same, i.e. y₀'=y₁'. In addition, if a line goes through two points, (x₀, y₀) and (x₁, y₁), the slope is
\(\displaystyle \frac{y_1 - y_0}{x_1 - x_0}\)
Thus we have
\(\displaystyle y_0' = y_1' = \frac{y_1 - y_0}{x_1 - x_0}\)
I believe that is enough, along with the equations for the circles, to compute the (set of) value(s) for (x₀, y₀) and (x₁, y₁)

NOTE: It may be better in some cases to work with the reciprocals.

 

[shandymilo]

Thank you Ill let you know how to goes

 

[Steven G]

That they are equal one another?

That would mean that this point is a point of intersection. Do you need derivatives to find a point of intersection?