Binomial Theorem

Question
CBSEENMA11015210

Verify that the locus of a point which moves so that the difference of its distances from points A (4, 0) and B (– 4, 0) is always equal to 2, is a hyperbola in the standard form <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)
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Solution

Let the point be straight P space left parenthesis straight alpha comma space straight beta right parenthesis.
The fixed points are A (4, 0), B (-4, 0). The given condition is :
                           PA - PB = 2
rightwards double arrow       <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)
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Squaring both sides, we get
                   space space straight alpha squared plus straight beta squared minus 8 straight alpha plus 16 space equals space 4 plus straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 plus 4 square root of straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 end root
rightwards double arrow                              negative 16 straight alpha minus 4 space equals space 4 square root of straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 end root space rightwards double arrow space minus left parenthesis 4 straight alpha plus 1 right parenthesis space equals space square root of straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 end root
Squaring again, we get
                        16 straight alpha squared plus 8 straight alpha plus 1 space equals space straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 space rightwards double arrow space 15 straight alpha squared minus straight beta squared space equals space 15 space rightwards double arrow space straight alpha squared over 1 minus straight beta squared over 15 equals 1
Hence, the locus of point P is straight x squared over 1 minus straight y squared over 15 equals 1
which is the standard form of hyperbola in the form straight x squared over straight a squared minus straight y squared over straight b squared equals 1.

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