Question

Three circles, whose radii are \[a,b,c\] touch each other externally and the tangents at their points of contact meet in a point. Then, the distance of this point from either point of contact is
A.\[\sqrt {\dfrac{{abc}}{{a + b + c}}} \]
B.\[\sqrt {\dfrac{{abc}}{{a - b - c}}} \]
C.\[2\sqrt {\dfrac{{abc}}{{a + b + c}}} \]
D.None of these

Answer 1

Hint: In the given question, we have been given three circles with given radii. They touch each other externally and their tangents meet at a point. We have to calculate the distance of this meeting point from either point of contact. To solve this, first we are going to need to draw the figure of the configuration. This problem involves using the formula of radius of the incenter circle of the triangle. We name the point of contact of the circles, join them along their radius, and form a triangle. Then we apply the required formula.

Formula Used:
We are going to use the formula of radius of inscribed circle, which is:
$r=\dfrac{\text{area of triangle}}{\text{semi perimeter of triangle}}$

Complete step-by-step answer:
First, we are going to draw the figure of the configuration and do the naming as below.

Now, \[ID,IE,IF\] are the common tangents meeting at the center point \[I\], so we have,
\[IE = ID = IF\]
Now, \[ID \bot BC\], \[IF \bot AB\] and \[IE \bot CA\], hence,
\[I\] is the incenter of \[\Delta ABC\] whose sides are given by \[a + b,b + c,c + a\].
Now, semi-perimeter of \[\Delta ABC\], \[s = \dfrac{{a + b + b + c + c + a}}{2} = a + b + c\]
Now, area of \[\Delta ABC\], \[A = \sqrt {s\left( {s - \left( {a + b} \right)} \right)\left( {s - \left( {b + c} \right)} \right)\left( {s - \left( {c + a} \right)} \right)} = \sqrt {sabc} \]
From the formula, \[r = \dfrac{{\sqrt {sabc} }}{s} = \dfrac{{\sqrt s .\sqrt {abc} }}{{{{\left( {\sqrt s } \right)}^2}}} = \dfrac{{\sqrt {abc} }}{{\sqrt s }} = \sqrt {\dfrac{{abc}}{{a + b + c}}} \]
Hence, the correct option is A

Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we use the formula describing the relation between these two things. It is really helpful if we first draw a sketch of the given scenario, as it helps in simply using the given information and not having to think of them.