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Conic Section

Short Answer Type

11. Find the equation of a circle, the two ends of whose diameter are A(4, –5) and R (–2, –1).

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Long Answer Type

12. Find the equation of a circle which passes through the points (2, 3) and (–1, 1) and whose centre lies on the straight line x – 3y – 11 = 0.

Let the equation of the circle be $space space space space left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared space equals space straight r squared$ ...(i)
It passes through (2, 3) $rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ ...(ii)
It passes through (-1, 1) $rightwards double arrow$ $left parenthesis negative 1 minus straight h right parenthesis squared plus left parenthesis 1 minus straight k right parenthesis squared equals straight r squared space rightwards double arrow space straight h squared plus straight k squared plus 2 straight h minus 2 straight k plus 2 equals straight r squared$ ...(iii)
From (ii) and (iii), we have
 $straight h squared plus straight k squared minus 4 straight h minus 6 straight k plus 13 equals straight h squared plus straight k squared plus 2 straight h minus 2 straight k plus 2$
$rightwards double arrow$ $negative 6 straight h minus 4 straight k plus 11 equals 0 space rightwards double arrow space 6 straight h plus 4 straight k minus 11 equals 0$ ...(iv)
Also, centre (h, k) lies on the line x - 3y -11 = 0 $rightwards double arrow$ h - 3k - 11 = 0 ...(v)
Multiplying (v) by 6, we get
6h - 18k - 66 =0 ...(vi)
Subtracting (vi) from (iv), we get $space space space space 22 straight k plus 55 equals 0 space rightwards double arrow space straight k equals fraction numerator negative 5 over denominator 2 end fraction$
∴ from (v), $space space space straight h minus 3 open parentheses negative 5 over 2 close parentheses minus 11 equals 0 space rightwards double arrow space straight h equals fraction numerator negative 15 over denominator 2 end fraction plus 11 equals 7 over 2$
Also, from (ii), $space space space 49 over 4 plus 25 over 4 minus 4 open parentheses 7 over 2 close parentheses minus 6 open parentheses negative 5 over 2 close parentheses plus 13 equals straight r squared$
$rightwards double arrow$ $straight r squared equals 49 over 4 plus 25 over 4 minus 14 plus 15 plus 13 space rightwards double arrow space space straight r squared equals 49 over 4 plus 25 over 4 plus 14$
$rightwards double arrow$ $straight r squared equals fraction numerator 49 plus 25 plus 56 over denominator 4 end fraction equals 130 over 4 equals 65 over 2$
Hence, the equation of circle is: $open parentheses straight x minus 7 over 2 close parentheses squared plus open parentheses straight y plus 5 over 2 close parentheses squared space equals space 65 over 2$
or $straight x squared plus straight y squared minus 7 straight x plus 5 straight y plus 49 over 4 plus 25 over 4 minus 65 over 2 equals 0$ or $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
or $space space straight x squared plus straight y squared minus 7 straight x plus 5 straight y minus 56 over 4 equals 0$
Hence, the equation of circle is: $space space straight x squared plus straight y squared minus 7 straight x plus 5 straight y minus 14 equals 0$