question_answer

Given inside a circle, whose radius is equal to 13 cm, is a point M at a distance 5 cm from the centre of the circle. A chord AB = 25 cm is drawn through M. The lengths of the segments into which the chord AB is divided by the point M in CM are

A)  12, 13

B)  14, 11

C)  15, 10

D)  16, 9  

Correct Answer: D

Solution :

 Draw a perpendicular OD from O to AB.                 \[\therefore \]  \[AD=\frac{25}{2}=12.5\] In right angled \[\Delta \,\,ODA,\]                 \[O{{A}^{2}}=O{{D}^{2}}+A{{D}^{2}}\]                 \[{{(13)}^{2}}=O{{D}^{2}}+{{(12.5)}^{2}}\] or            \[O{{D}^{2}}=169-156.25\]                 \[=12.75\] Again in right angled \[\Delta \,ODM,\]                 \[O{{M}^{2}}=O{{D}^{2}}+D{{M}^{2}}\]                 \[{{5}^{2}}=12.75+D{{M}^{2}}\] \[\therefore \]      \[D{{M}^{2}}=25-12.75=12.55\]                 \[DM=\sqrt{12.25}=3.5\,cm\] \[\therefore \]  \[AM=12.5+3.5=16\,cm\] and        \[MB=9\,cm\]