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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 = 2\sqrt{2}and C2 ŗ (5, 5), r2 =50λ\sqrt{50 - \lambda}

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. 50λ\sqrt{50 - \lambda}2\sqrt{2} < 32+32\sqrt{3^{2} + 3^{2}} < 50λ\sqrt{50 - \lambda} + 2\sqrt{2}

i.e. 50λ\sqrt{50 - \lambda} < 42\sqrt{2} and 50λ\sqrt{50 - \lambda} > 222\sqrt{2}

gives l > 18 and l < 42

Hence, we have 18 < l < 42.