The set of all real values of X for which exactly two common tangents can be drawn to the circles x2+y2-4x-4y+6 = 0 and x2+y2-10x-10y+λ = 0 is the interval :

Question

The set of all real values of X for which exactly two common tangents can be drawn to the circles x2+y2-4x-4y+6 = 0 and x2+y2-10x-10y+λ = 0 is the interval : (A) (12, 32) (B) (18, 42) (C) (12, 24) (D) (18, 48)

Tardigrade Admin · Accepted Answer

C1 (2, 2)C2 (5, 5) r1 = √2 r2 = √50-1 |r1-r2| < c1 c2 < r1+r2 |√50-λ-√2|< √9+9 < √50-λ +√2 -18 <[√50-λ -√2] < 18 √18-√2 < √50-λ λ > 18 20-12 < 50-λ λ < 42 λ ∈ (18, 42)

Q. The set of all real values of X for which exactly two common tangents can be drawn to the circles
$x^{2} + y^{2} - 4 x - 4 y + 6 = 0$ and
$x^{2} + y^{2} - 10 x - 10 y + λ = 0$ is the interval :

(12, 32)

(18, 42)

(12, 24)

(18, 48)

$C_{1} (2, 2) C_{2} (5, 5)$
$r_{1} = 2 r_{2} = 50 - 1$
$∣ r_{1} - r_{2} ∣ < c_{1} c_{2} < r_{1} + r_{2}$
$∣ ∣ 50 - λ - 2 ∣ ∣ < 9 + 9 < 50 - λ + 2$
$- 18 < [50 - λ - 2] < 1818 - 2 < 50 - λ$
$λ > 1820 - 12 < 50 - λ$
$λ < 42$
$λ \in (18, 42)$

Q. The set of all real values of X for which exactly two common tangents can be drawn to the circles x2+y2−4x−4y+6=0 and x2+y2−10x−10y+λ=0 is the interval :