Find the range of r for which the circles \({\left( {x - 1} \right)^2} + {\left( {y - 3} \right)^2} = {r^2}\;and\;{\left( {x - 1} \right)^2} + {\left( {y + 1} \right)^2} = 9\) intersect at two different points?

Option 4 : (1, 7)

Equation of circle,

Where (h, k) is centre of the circle and r is the radius of the circle.

For this equation,

h = 1 and k = 3, therefore centre is C₁(1, 3)

For another equation,

h = 1 and k = 1, therefore centre is C₂(1, -1) and radius r = 3

The distance between two centres is

So if r = 1, there will be a single point of contact between C₁ and C₂.

there will be a single point of contact between C₁ and C₂.

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Hence the range of r is (1, 7).