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Conic Section

Short Answer Type

111. Find the equation of an ellipse for which foci are at (± 3, 0), a = 4 and centre is at origin. Also, find the eccentricity of the ellipse.

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112. Find the equation of an ellipse for which c = 4, b = 3, centre is at the origin and foci on x-axis. Determine the eccentricity of the ellipse.

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Long Answer Type

113.
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 $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

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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114.

For ellipse $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ verify $PS subscript 1 plus PS subscript 2 equals 10 comma$ where $space straight S subscript 1$ and $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ be the foci of the ellipse and P is any point on it.

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115. A rod of length 9 cm moves with its ends always touching the co-ordinate axes. Show that the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis, is an ellipse.

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116. A man running a racecourse notes that the sum of the distances from two flag posts is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

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Short Answer Type

117. The arch given below is in form of a semi-ellipse. It is 8 m wide and its height at the centre is 2 m. Find the height of the arch at a distance of 1.5 m from one end.

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Long Answer Type

118.

In the following, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$straight x squared over 16 plus straight y squared over 9 equals 1$