Question

The set of all real values of \(\lambda\) for which exactly two common tangents can be drawn to the circles \(x^2 + y^2 - 4x - 4y + 6 = 0\) and \(x^2 + y^2 - 10x - 10y + \lambda = 0\) is the interval:

• A. \((12, 32)\)
• B. \((18, 42)\)
• C. \((12, 24)\)
• D. \((18, 48)\)

Hint:

Exactly two common tangents means the circles intersect at two points: \(|r_1 - r_2| \lt d \lt r_1 + r_2\).

Answer 1

The Correct Option is B

Circle 1: centre \(C_1(2,2)\), \(r_1 = \sqrt{4+4-6} = \sqrt2\).
Circle 2: centre \(C_2(5,5)\), \(r_2 = \sqrt{50 - \lambda}\) (needs \(\lambda \lt 50\)).
\(d = C_1C_2 = 3\sqrt2\).
Exactly two common tangents \(\Leftrightarrow\) \(|r_1 - r_2| \lt d \lt r_1 + r_2\).
\(d \lt r_1 + r_2 \Rightarrow r_2 \gt 2\sqrt2 \Rightarrow 50 - \lambda \gt 8 \Rightarrow \lambda \lt 42\).
\(|r_2 - r_1| \lt d \Rightarrow r_2 \lt 4\sqrt2 \Rightarrow 50 - \lambda \lt 32 \Rightarrow \lambda \gt 18\).
Hence \(\lambda \in (18, 42)\).
Answer: (2) \((18, 42)\)