Question

Two circles of radius 1 and 4 touch each other externally. Another circle of radius r touches both circles externally and also one direct common tangent of the two circles. Then `[1/r]=`

Answer 1

`In/_BFO`
`OB^2=BF^2+OF^2`
`(r+4)^2=(4-r)^2+(d-x)^2`
`(d-r)^2=(x+4)^2-(4-r)^2`
`=r^2+16+8r-16-r^2+8r`
`=16r`
`d-x=4sqrtr-(1)`
`In/_OCE`
`OC^2=CE^2+OE^2`
`(r+1)^2=(1-r)^2+x^2`
`x^2=(r+1)^2-(1-r)^2`
`x=2sqrtr-(2)`
`In/_BGC`
`(4+1)^2=(4-1)^2+d^2`
`d^2=(4+1)^2-(4-1)^2=25-9=16`
`d=4`
`d-x=4sqrtr`
`4-x=4sqrtr`
`4-2sqrtr=4sqrtr`
`4=6sqrtr`
`sqrtr=2/3`
`r=4/9`
`[1/r]=[9/4]=2`.