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Question

# Consider the following statement : I. Two circles with centres A and B, radii 3 cm and 4 cm respectively intersect at two points C and D. II. AC and BC are tangents to the two circles. Then, length of chord CD will be___. Choose the correct option.

A
Statement I alone is sufficient to answer
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B
Statement II alone is sufficient to answer
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C
Both statements are required to answer
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D
Neither of the statement is sufficient.
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Solution

## The correct option is C Both statements are required to answerGiven, two circles with centres A and B, AC and BC are two tangents. ∴ Radius of circle (i) =AC =3 cm and radius of circle (ii) =BC =4 cm In ΔACB, AC⊥BC Since, AC is radius and BC is tangent ∴AC2+BC2=AB2⇒32+42=AB2⇒AB=5cm Since, the line joining the centre of two intersecting circle is perpendicular bisector of their common chord. Let BM=x cm ∴ In ΔBMC, BM⊥CM∴BC2=CM2+BM2⇒42=CM2+x2⇒42−x2=CM2⇒√16−x2=CM Also, in ΔACM, AC2=CM2+AM2⇒32=(16−x2)+(5−x)2⇒32=16−x2+25+x2−10x⇒9=16+25−10x⇒−32=−10xx=3.2 In ΔCMB, CM=√16−(3.2)2=2.4cm CD=2CM=4.8cm Hence, both statements are required

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