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  • A triangle is formed by the lines  and the two common tangents of the circles

A triangle is formed by the lines \mathrm{x+y=1} and the two common tangents of the circles \mathrm{x^2+y^2-2 x=0\, \, and \, \, x^2+y^2-4 x-2 y+1=0}. Its area is ________.

Option: 1

3


Option: 2

4


Option: 3

2


Option: 4

7


Answers (1)

best_answer

The centres of the circles C_1(1,0) and C_2(2,1) and radii are r_1=1, r_2=2. The common tangents meet at T which divides C_1 C_2 in the ratio -1: 2.

\therefore \quad T \equiv(0,-1)
\Rightarrow The pair of tangents from T is given by

\begin{aligned} & S_1{ }^2=S S_{11} \\ & (-y-x)^2=\left(x^2+y^2-2 x\right) 1 \\ \Rightarrow & x(y+1)=0 \end{aligned}

The sides of the triangle are x+y=1,

x=0, y+1=0 \Rightarrow \text { Area }=\frac{1}{2} \cdot 2 \cdot 2=2 .

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