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

Answer 1

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.

$\therefore$ Radius of circle (i) =AC =3 cm and radius of circle (ii) =BC =4 cm
In $\Delta ACB,$
$AC\perp BC$
Since, AC is radius and BC is tangent
$\therefore A{C}^2+B{C}^2=A{B}^2\Rightarrow 3^2+4^2=A{B}^2\Rightarrow AB=5cm$
Since, the line joining the centre of two intersecting circle is perpendicular bisector of their common chord.
Let BM$=xcm$
$\therefore$ In $\Delta BMC,$
$BM\perp CM\therefore B{C}^2=C{M}^2+B{M}^2\Rightarrow 4^2=C{M}^2+x^2\Rightarrow 4^2-x^2=C{M}^2\Rightarrow \sqrt{16-x^2}=CM$
Also, in $\Delta ACM,$
$A{C}^2=C{M}^2+A{M}^2\Rightarrow 3^2=(16-x^2)+(5-x{)}^2\Rightarrow 3^2=16-x^2+25+x^2-10x\Rightarrow 9=16+25-10x\Rightarrow -32=-10xx=3.2$
In $\Delta CMB,$
$CM=\sqrt{16-(3.2{)}^2}=2.4cm$
$CD=2CM=4.8cm$

Hence, both statements are required