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Conic Section

Short Answer Type

91. Find equation of parabola for vertex at origin, focus (0, –2), directrix y = 2.

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92. Find equation of parabola for vertex at origin, focus (–3, 0), directrix x = 3.

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93. Find equation of parabola for vertex at origin, axis along the y-axis and passing through the point (3, 2).

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94. Find equation of parabola for vertex at the origin, symmetric with respect to the x-axis and passing through point (2, –3).

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Long Answer Type

95.

The cable of a uniformly loaded suspension bridge hang in the form of a parabola.
The roadway which is horizontal and 100 m long, is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to a roadway 18 m from the middle.

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96. Verify that the locus of a point P which moves so that the sum of its distance from points S₁ (– 4,0) and S₂ (4,0) is 10, is an ellipse.

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97. In the following, find the co-ordinates 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 36 plus straight y squared over 16 equals 1

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98. In the following, find the co-ordinates 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.

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99. In the following equation, Find the co-ordinates of foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of latus rectum of the ellipse.

16 straight x squared plus straight y squared equals 16

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100. In the following equation, Find the co-ordinates of foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of latus rectum of the ellipse.
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Equation of ellipse is : $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow$ $fraction numerator 4 straight x squared over denominator 36 end fraction plus fraction numerator 9 straight y squared over denominator 36 end fraction equals 1 space space space rightwards double arrow space space straight x squared over 9 plus straight y squared over 4 equals 1$
Now, $9 greater than 4 space rightwards double arrow space straight a squared equals 9 comma space straight b squared equals 4 space rightwards double arrow space straight a equals 3 comma space straight b equals 2$
∴ Equation of ellipse in the standard form is $space space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1$
(i) Focus, $straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space open parentheses negative square root of straight a squared minus straight b squared end root comma space 0 close parentheses space left right arrow space left parenthesis negative square root of 9 minus 4 end root comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of 5 comma space 0 right parenthesis$
Focus, $straight S subscript 2 space left parenthesis straight c comma space 0 right parenthesis space left right arrow space left parenthesis square root of 5 comma space 0 right parenthesis$

(ii) Vertex, $straight A space left parenthesis straight a comma space 0 right parenthesis space left right arrow space left parenthesis 3 comma space 0 right parenthesis$
Vertex, $straight A apostrophe left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space left parenthesis negative 3 comma space 0 right parenthesis$

(iii) Length of major axis = 2a = 2(3) = 6
Length of minor axis = 2b = 2(2) = 4

(iv) Let the eccentricity be e.
Now, c = ae, $rightwards double arrow$ $square root of 5 space equals space 3 straight e$ ∴ $straight e equals fraction numerator square root of 5 over denominator 3 end fraction$
(v) Length of latus rectum $equals space fraction numerator 2 straight b squared over denominator straight a end fraction space equals space fraction numerator 2 left parenthesis 4 right parenthesis over denominator 3 end fraction equals 8 over 3$

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