Question
Question: If exactly two common tangents can be drawn to the circles x<sup>2</sup> + y<sup>2</sup> – 4x – 4y ...
If exactly two common tangents can be drawn to the circles
x2 + y2 – 4x – 4y + 6 = 0 and x2 + y2 – 10x – 10y + l = 0, then
A
l< 32
B
18 <l< 42
C
l< 24
D
0 <l< 24
Answer
18 <l< 42
Explanation
Solution
We have
C1 ŗ (2, 2), r = 2and C2 ŗ (5, 5), r2 =50−λ
For only two real common tangents, the two circles must
intersect in two real distinct points. Thus, we have
|r1 – r2| < C1C2 < |r1 + r2|
i.e. 50−λ– 2 < 32+32 < 50−λ + 2
i.e. 50−λ < 42 and 50−λ > 22
gives l > 18 and l < 42
Hence, we have 18 < l < 42.