Quantitative Ability Question of the Day

The equations of two circles are
x^2+y^2=R^2 and (x – 4)^2 + (y + 3)^2 = 1.
What is the condition on R for the circles to have exactly two common tangents?


OPTIONS
1) R < 5
2) R > 4
3) 3 < R < 4
4) 4 < R < 6

SOLUTION

For two circles to have exactly one common tangent, they must touch internally, which means that the distance between their centres is the difference of their radii.

For two circles to have exactly three common tangents, they must touch externally, which means that the distance between their centres is the sum of their radii.
If the distance between their centres is between the sum and difference of their radii, they will have exactly two common tangents.
The centres of the given circles are (0, 0) and (4, –3).
The distance between these centres is

The radii of the given circles are R and 1.
∴ The sum of their radii is (R + 1), and the difference is (R – 1).
The condition for them to have exactly two common tangents is
R – 1 < 5 < R + 1,
which gives 4 < R < 6.
Hence, option 4.