Binomial Theorem

Question
CBSEENMA11015178

Let P be any point on the ellipse space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1 with the two foci at space straight S subscript 1 and straight S subscript 2. Verify that <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>

Solution

Let point P be (x, y). Focus space space straight S subscript 1 is (-c, 0),  <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> is (c, 0)
rightwards double arrow                       PS subscript 1 equals square root of left parenthesis straight x plus straight c right parenthesis squared plus straight y squared end root comma space PS subscript 2 equals square root of left parenthesis straight x minus straight c right parenthesis squared plus straight y squared end root
∴        space space PS subscript 1 plus PS subscript 2 space equals space square root of left parenthesis straight x plus straight c right parenthesis squared plus straight y squared end root space plus space square root of left parenthesis straight x minus straight c right parenthesis squared plus straight y squared end root
Now,          straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space space space rightwards double arrow space straight y squared over straight b squared equals 1 minus straight x squared over straight a squared equals fraction numerator straight a squared minus straight x squared over denominator straight a squared end fraction rightwards double arrow space straight y squared equals straight b squared open parentheses fraction numerator straight a squared minus straight x squared over denominator straight a squared end fraction close parentheses
But,                          straight c squared equals straight a squared minus straight b squared space rightwards double arrow space straight b squared equals straight a squared minus straight c squared
∴                             straight y squared equals open parentheses fraction numerator straight a squared minus straight c squared over denominator straight a squared end fraction close parentheses left parenthesis straight a squared minus straight x squared right parenthesis
∴   PS subscript 1 plus PS subscript 2 space equals space square root of left parenthesis straight x plus straight c right parenthesis squared plus open parentheses fraction numerator straight a squared minus straight c squared over denominator straight a squared end fraction close parentheses left parenthesis straight a squared minus straight x squared right parenthesis end root plus square root of left parenthesis straight x minus straight c right parenthesis squared plus open parentheses fraction numerator straight a squared minus straight c squared over denominator straight a squared end fraction close parentheses left parenthesis straight a squared minus straight x squared right parenthesis end root
<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>
   equals space 1 over straight a open square brackets square root of straight a to the power of 4 plus 2 straight a squared xc plus straight c squared straight x squared end root plus square root of straight a to the power of 4 minus 2 straight a squared xc plus straight c squared straight x squared end root close square brackets space equals space 1 over straight a open square brackets square root of left parenthesis straight a squared plus cx right parenthesis squared end root plus square root of left parenthesis straight a squared minus cx right parenthesis squared end root close square brackets
    equals space 1 over straight a left square bracket straight a squared plus cx plus straight a squared minus cx right square bracket space equals space 1 half left square bracket 2 straight a squared right square bracket space equals space 2 straight a
Hence, PS subscript 1 plus PS subscript 2 space equals space 2 straight a, which verifies the ellipse property, space space PS subscript 1 plus PS subscript 2 equals 2 straight a

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