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  • A circle C of unit radius lies in the first quadrant and touches both the axes. The circle t

A circle C of unit radius lies in the first quadrant and touches both the axes. The circle \mathrm{C_1} touches both the axes and intersects C such that the common chord is longest. Its radius is

Option: 1

\frac{1}{3}


Option: 2

3


Option: 3

\mathrm{\text { Both (a) and (b) }}


Option: 4

2


Answers (1)

best_answer

\mathrm{ \begin{aligned} & \text { (c) : } C: x^2+y^2-2(x+y)+1=0, \\\\ & C_1: x^2+y^2-2 r(x+y)+r^2=0 \end{aligned} }
The common chord is \mathrm{C-C_1=0}

\mathrm{ \Rightarrow 2(r-1)(x+y)=r^2-1 }

or \mathrm{2(x+y)=1+r}                                   ....(i)

If \mathrm{r<1,(r, r)} lies on (i)

\mathrm{ \therefore 2(r+r)=1+r \Rightarrow r=\frac{1}{3} }
If  \mathrm{r>1,(1,1)} lies on (i)

\mathrm{ \begin{aligned} & \therefore \quad 2(1+1)=1+r \\ & \Rightarrow r=3 . \end{aligned} }

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vishal kumar

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