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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 answer
Given, 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,
ACBC
Since, AC is radius and BC is tangent
AC2+BC2=AB232+42=AB2AB=5cm
Since, the line joining the centre of two intersecting circle is perpendicular bisector of their common chord.
Let BM=x cm
In ΔBMC,
BMCMBC2=CM2+BM242=CM2+x242x2=CM216x2=CM
Also, in ΔACM,
AC2=CM2+AM232=(16x2)+(5x)232=16x2+25+x210x9=16+2510x32=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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