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Conic Section

Long Answer Type

101.
Find: (i) the co-ordinates of foci.
(ii) the co-ordinates of vertices.
(iii) the length of major axis and minor axis.
(iv) the eccentricity.
(v) latus rectum of ellipse.
$4 straight x squared plus 9 straight y squared equals 1$

Equation of ellipse is : $space space space 4 straight x squared plus 9 straight y squared equals 1 space space space rightwards double arrow space space fraction numerator straight x squared over denominator open parentheses begin display style 1 fourth end style close parentheses end fraction plus fraction numerator straight y squared over denominator open parentheses begin display style 1 over 9 end style close parentheses end fraction equals 1$
Now, $1 fourth greater than 1 over 9 space rightwards double arrow space space straight a squared equals 1 fourth comma space straight b squared equals 1 over 9$
The equation of the ellipse in standard form is $space space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1$
 $straight a squared equals 1 fourth space rightwards double arrow space straight a equals 1 half space and space straight b squared equals 1 over 9 space space rightwards double arrow space straight b equals 1 third$
Now, $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
(i) Focus, $straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space open parentheses negative fraction numerator square root of 5 over denominator 6 end fraction comma 0 close parentheses$
Focus, $straight S subscript 2 left parenthesis straight c comma space 0 right parenthesis space left right arrow space open parentheses fraction numerator square root of 5 over denominator 6 end fraction comma 0 close parentheses$
(ii) Vertex, $straight A space left parenthesis straight a comma space 0 right parenthesis space left right arrow space open parentheses 1 half comma space 0 close parentheses$
Vertex, $straight A apostrophe left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space open parentheses negative 1 half comma space 0 close parentheses$
(iii) Length of major axis = 2a = $2 open parentheses 1 half close parentheses space equals space 1$
Length of minor axis = 2b = $2 open parentheses 1 third close parentheses equals 2 over 3$
(iv) Let the eccentricity be e
Now, c = ae $rightwards double arrow$ $fraction numerator square root of 5 over denominator 6 end fraction equals 1 half straight e space rightwards double arrow space space space straight e equals fraction numerator square root of 5 over denominator 6 end fraction cross times 2 over 1 space rightwards double arrow space straight e equals fraction numerator square root of 5 over denominator 3 end fraction$
(v) Length of latus rectum = $2 straight b squared over straight a equals fraction numerator 2 cross times 1 cross times 2 over denominator 9 cross times 1 end fraction equals 4 over 9$

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102.

Find: (i) the co-ordinates of foci.
(ii) the co-ordinates of vertices.
(iii) the length of major axis and minor axis.
(iv) the eccentricity.
(v) latus rectum of ellipse.
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Short Answer Type

103. Find the equation of the ellipse that satisfies the following condition:
Vertices (± 6, 0), foci (± 4, 0).

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104. Find the equation of the ellipse that satisfies the following condition:
Vertices (0, ± 13), foci (0, ± 5).

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105. Find the equation of an ellipse that satisfies the following conditions
Ends of major axis at

<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>

, ends of minor axis at

left parenthesis 0 comma space plus-or-minus 2 right parenthesis

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106. Find the equation of an ellipse that satisfies the following conditions
Ends of major axis at

space left parenthesis 0 comma space plus-or-minus square root of 13 right parenthesis comma

ends of minor axis at

space space left parenthesis plus-or-minus 2 comma space 0 right parenthesis

. Also find the co-ordinate of foci.

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107. Find the equation of an ellipse that satisfy the following condition:
Length of major axis 26, foci (± 5, 0).

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108. Find the equation of an ellipse that satisfy the following condition:
Length of minor axis 16, foci (0, ± 6). Find also, the eccentricity .

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109. Find the equation of an ellipse whose major axis is along the x-axis and which passes through the points (6, 3) and (12, 0).

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110. Find the equation of an ellipse whose major axis is along the y-axis and which passes through the points P (2, 2) and Q (1, 4).

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