There are two circles whose equation are `x^2+y^2=9` and `x^2+y^2-8x-6y+n^2=0,n in Zdot` If the two circles have exactly two common tangents, then the number of possible values of `n` is 2 (b) 8 (c) 9 (d) none of these

Options

A. 2
B. 8
C. 9
D. none of these

Answer

C

Explanation

The coordinates of the centres and radii of the circles are:
`{:("Centres:",C_(1)(0, 0),C_(2)(4, 3)),("Radii",r_(1)=3,r_(2)=sqrt(25-n^(2))-5 lt n lt 5),(,,):}`
Given circles will have exactly two common tangents, if
`|r_(1)-r_(2)| lt C_(1) C_(2) lt R_(1)+r_(2)`
`rArr |3-sqrt(25-n^(2))| lt 5` is true for all `n in (-5, 5)`.
Now,
`5 lt 3 + sqrt(25-n^(2)) `
`rArr 2 lt sqrt(25-n^(2))`
`rArr 4 lt 25 - n^(2)`
`rArr n^(2)-21 lt 0 `
`rArr -sqrt (21) lt n lt sqrt(21) rArr n = pm 4, pm 3, pm 2, pm1, 0`.
Hence, n can take 9 integral values.