Question

Two circles c₁: x² +  y² - 4x - 6y - 3 = 0 and c₂: x² + y² + 2x - 14y + λ meet at two distinct points then find the value of λ.

(1) -31 < λ < 40

(2) -31 < λ < 49

(3) (-20 < λ < 49)

(4) (-11 < λ < 49)

Answer 1

Correct option is (2) -31 < λ < 49

c₁ = (2, 3),  r₁ = 4

c₂ = (-1, 7), r₂ = \(\sqrt{50 - \lambda}\)

c₁c₂ = 5

\(|r_1 - r_2| <c_1 c_2 < r_1 + r_2\)

\(|4 - \sqrt{50 - \lambda}| < 5 < 4 + \sqrt{50 - \lambda}\)

\(\Rightarrow 1 < \sqrt{50 - \lambda} \Rightarrow \lambda < 49\)

\(\Rightarrow -5 < \sqrt{50 - \lambda} - 4 < 5\)

\(\Rightarrow \sqrt{50 - \lambda} < 9 \)

\(\Rightarrow 50 -\lambda< 81\)

\(\Rightarrow -31< \lambda\)

\(\Rightarrow -31< \lambda <49\)