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Let point P be (x, y). Focus is (-c, 0), is (c, 0) ∴ Now, But, ∴ ∴ Hence, , which verifies the ellipse property, - Wired Faculty

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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 $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

Solution

Let point P be (x, y). Focus $space space straight S subscript 1$ is (-c, 0), $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ 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, $straight c squared equals straight a squared minus straight b squared space rightwards double arrow space straight b squared equals straight a squared minus straight c squared$
∴ $straight y squared equals open parentheses fraction numerator straight a squared minus straight c squared over denominator straight a squared end fraction close parentheses left parenthesis straight a squared minus straight x squared right parenthesis$
∴ $PS subscript 1 plus PS subscript 2 space equals space square root of left parenthesis straight x plus straight c right parenthesis squared plus open parentheses fraction numerator straight a squared minus straight c squared over denominator straight a squared end fraction close parentheses left parenthesis straight a squared minus straight x squared right parenthesis end root plus square root of left parenthesis straight x minus straight c right parenthesis squared plus open parentheses fraction numerator straight a squared minus straight c squared over denominator straight a squared end fraction close parentheses left parenthesis straight a squared minus straight x squared right parenthesis end root$
$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$equals space 1 over straight a open square brackets square root of straight a to the power of 4 plus 2 straight a squared xc plus straight c squared straight x squared end root plus square root of straight a to the power of 4 minus 2 straight a squared xc plus straight c squared straight x squared end root close square brackets space equals space 1 over straight a open square brackets square root of left parenthesis straight a squared plus cx right parenthesis squared end root plus square root of left parenthesis straight a squared minus cx right parenthesis squared end root close square brackets$
 $equals space 1 over straight a left square bracket straight a squared plus cx plus straight a squared minus cx right square bracket space equals space 1 half left square bracket 2 straight a squared right square bracket space equals space 2 straight a$
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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