question_answer

AB and CD are two parallel chords of a circle such that AB = 10 cm and CD = 24 cm. If the chords are on the opposite sides of the centre and distances between them is 17 cm, then the radius of the circle is

A) 12 cm              

B) 13 cm

C) 10 cm              

D) 11 cm

Correct Answer: B

Solution :

Let radius of the circle be r.

Given,   \[MN=17\,cm\]

Let        \[ON=x\]

Then,    \[OM=17-x\]

In \[\Delta AOM,\]\[O{{A}^{2}}=O{{M}^{2}}+A{{M}^{2}}\]

\[\Rightarrow \]   \[{{r}^{2}}={{(17-x)}^{2}}+{{5}^{2}}\]

\[=289+{{x}^{2}}-34x+25\]

\[{{r}^{2}}=314+{{x}^{2}}-34x\]                 ... (i)

In \[\Delta CON,\]\[O{{C}^{2}}=O{{N}^{2}}+C{{N}^{2}}\]

\[{{r}^{2}}={{x}^{2}}+{{12}^{2}}={{x}^{2}}+144\]                  ... (ii)

From Eqs. (i) and (ii), we get

\[314+{{x}^{2}}-34x={{x}^{2}}+144\]

\[\Rightarrow \]               \[34x=170\]\[\Rightarrow \]\[x=5\]

From Eq. (ii),

\[{{r}^{2}}=25+144=169\]

\[\Rightarrow \]   \[r=13\,cm\]