Binomial Theorem

Question
CBSEENMA11015077

Find the equation of a circle which passes through the points (2, 3) and (–1, 1) and whose centre lies on the straight line x – 3y – 11 = 0.

Solution

Let the equation of the circle be space space space space left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared space equals space straight r squared                                                                       ...(i)
It passes through (2, 3) rightwards double arrow  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>                             ...(ii)
It passes through (-1, 1) rightwards double arrow    left parenthesis negative 1 minus straight h right parenthesis squared plus left parenthesis 1 minus straight k right parenthesis squared equals straight r squared space rightwards double arrow space straight h squared plus straight k squared plus 2 straight h minus 2 straight k plus 2 equals straight r squared                            ...(iii)
From (ii) and (iii), we have
              straight h squared plus straight k squared minus 4 straight h minus 6 straight k plus 13 equals straight h squared plus straight k squared plus 2 straight h minus 2 straight k plus 2
rightwards double arrow                  negative 6 straight h minus 4 straight k plus 11 equals 0 space rightwards double arrow space 6 straight h plus 4 straight k minus 11 equals 0                                                                        ...(iv)
Also, centre (h, k) lies on the line x - 3y -11 = 0 rightwards double arrow h - 3k - 11 = 0                                                           ...(v)
Multiplying (v) by 6, we get 
                            6h - 18k - 66 =0                                                                                                        ...(vi)
Subtracting (vi) from (iv), we get space space space space 22 straight k plus 55 equals 0 space rightwards double arrow space straight k equals fraction numerator negative 5 over denominator 2 end fraction
∴ from (v), space space space straight h minus 3 open parentheses negative 5 over 2 close parentheses minus 11 equals 0 space rightwards double arrow space straight h equals fraction numerator negative 15 over denominator 2 end fraction plus 11 equals 7 over 2
Also, from (ii),   space space space 49 over 4 plus 25 over 4 minus 4 open parentheses 7 over 2 close parentheses minus 6 open parentheses negative 5 over 2 close parentheses plus 13 equals straight r squared
rightwards double arrow                            straight r squared equals 49 over 4 plus 25 over 4 minus 14 plus 15 plus 13 space rightwards double arrow space space straight r squared equals 49 over 4 plus 25 over 4 plus 14
rightwards double arrow                             straight r squared equals fraction numerator 49 plus 25 plus 56 over denominator 4 end fraction equals 130 over 4 equals 65 over 2
Hence, the equation of circle is: open parentheses straight x minus 7 over 2 close parentheses squared plus open parentheses straight y plus 5 over 2 close parentheses squared space equals space 65 over 2
or   straight x squared plus straight y squared minus 7 straight x plus 5 straight y plus 49 over 4 plus 25 over 4 minus 65 over 2 equals 0 or <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
or  space space straight x squared plus straight y squared minus 7 straight x plus 5 straight y minus 56 over 4 equals 0
Hence, the equation of circle is: space space straight x squared plus straight y squared minus 7 straight x plus 5 straight y minus 14 equals 0

Sponsor Area

Some More Questions From Binomial Theorem Chapter