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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.

Let AOB be the cable of a uniformly loaded suspension bridge as shown in the figure.
Let AL and BM be the longest wires of length 30 m each. Let OC be the shortest wire of length 6 m and LM be the road way as shown in the figure.

The longest supporting wire = AL = BM = 30 m
The shortest supporting cable = OC = 6 m
Length of the roadway = LM = 100 m
∴ LC = CM = 50 m
Let O be the vertex and OY be the axis of the parabola
∴ Equation of the parabola is $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ ...(i)
ON = CM = 50
BN = BM - MN = BM - OC = 30 - 6 = 24
∴ Co-ordinates of point B are (50, 24)
Since it lies on the parabola (i)
∴ $left parenthesis 50 right parenthesis squared equals 4 left parenthesis straight a right parenthesis space left parenthesis 24 right parenthesis space or space straight a equals 2500 over 96 equals 625 over 24$
∴ From (i), the equation of the parabola is: $straight x squared equals 4 open parentheses 625 over 24 close parentheses straight y$
or $straight x squared equals 625 over 6 straight y$ ...(ii)
Let the length of the supporting wire PQ at a distance of 18 m be h.
∴ OR = CQ = 18 m
PR = PQ - RQ = PQ - OC = h - 6
$rightwards double arrow$ Co-ordinates of point P are (18, h - 6)
Since it lies on parabola (ii)
∴ $space space open parentheses 18 close parentheses squared space equals space 625 over 6 left parenthesis straight h minus 6 right parenthesis$
$rightwards double arrow$ $space space space space 324 cross times 6 space equals space 625 space straight h space minus space 3750$
$rightwards double arrow$ 1944 + 3750 = 625 h
$rightwards double arrow$ $straight h equals 5694 over 625 equals 9.11 space straight m$

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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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