Question

# For the given circles x2+y2+2x+4yâˆ’20=0 and x2+y2+6xâˆ’8y+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 unitsGiven circles are x2+y2+2x+4y−20=0 and x2+y2+6x−8y+10=0 The centre and radius are, C1=(−1,−2), r1=√1+4+20=5C2=(−3,4), r2=√9+16−10=√15 Now, C1C2=√22+62=√40⇒r1+r2>C1C2>|r1−r2| Therefore, the circles intersect in two distinct points, so they have 2 common tangents. Now, the length of common tangent (DCT) =√(d)2−(r1−r2)2=√40−(5−√15)2 (∵d=C1C2)=(1500)1/4

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