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Question

For the given circles x2+y2+2x+4y20=0 and x2+y2+6x8y+10=0, which of the following is/are correct?

A
The number of common tangents is 2.
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B
The number of common tangents is 3.
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C
Length of the common tangent is (1500)1/4 units
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D
Length of the common tangent is (1000)1/4 units
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Solution

The correct options are
A The number of common tangents is 2.
C Length of the common tangent is (1500)1/4 units
Given circles are
x2+y2+2x+4y20=0 and x2+y2+6x8y+10=0
The centre and radius are,
C1=(1,2), r1=1+4+20=5C2=(3,4), r2=9+1610=15
Now,
C1C2=22+62=40r1+r2>C1C2>|r1r2|
Therefore, the circles intersect in two distinct points, so they have 2 common tangents.

Now, the length of common tangent (DCT)
=(d)2(r1r2)2=40(515)2 (d=C1C2)=(1500)1/4

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