Find the minimum radius of the circle which is orthogonal to two given circles

Question

Problem:

Find the minimum radius of the circle which is orthogonal to both the circles $x^2+y^2-12x+35=0$ and $x^2+y^2+4x+3=0$ .

Solution:

Let the equations : $x^2+y^2-12x+35=0\tag i$

and

$x^2+y^2+4x+3=0\tag {ii}$

Equation of radical axis of $(i)$ and $(ii)$ is $-16x +32=0 \Rightarrow x =2$

It intersects the line joining the center, i.e., $y =0$ at the point $(2,0)$.

Question:

How do I find the minimal radius of the circle?

Answer 1

Your approach was correct and you are indeed very close to your answer.

From the radical axis equation it is pretty obvious that the centre of the orthogonal circle will lie have coords (2,y).

Because the radius of the orthogonal circle will be the length of tangent to any one of the circle, hence , by finding the length of tangent we can arrive at our answer.

Using Length of Tangent(L) = √S1 , where S1 = the value we get after plugging in the coords of an external point

therefore L = √(y^2 + 15)

now to get get minimum length, you can either differentiate it wrt y or just plug in y=0.

hence minimum radius is √15