Question

Verify that the locus of a point which moves so that the difference of its distances from points A (0, –5) and B (0,5) is always 6, is a hyperbola in the standard form
$space space straight y squared over straight a squared minus straight x squared over straight b squared equals 1.$

Solution

Let the point be $straight P space left parenthesis straight alpha comma space straight beta right parenthesis.$ It is given that
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$rightwards double arrow$ $square root of straight alpha squared plus straight beta squared plus 10 straight beta plus 25 end root space equals space 6 plus square root of straight alpha squared plus straight beta squared minus 10 straight beta plus 25 end root$
Squaring both sides, we get
$straight alpha squared plus straight beta squared plus 10 straight beta plus 25 equals 36 plus straight alpha squared plus straight beta squared minus 10 straight beta plus 25 plus 12 square root of straight alpha squared plus straight beta squared minus 10 straight beta plus 25 end root$
$rightwards double arrow$ $20 straight beta minus 36 space equals space 12 square root of straight alpha squared plus straight beta squared minus 10 straight beta plus 25 end root space space rightwards double arrow space 5 straight beta minus 9 space equals space 3 square root of straight alpha squared plus straight beta squared minus 10 straight beta plus 25 end root$
Squaring again, we get
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$rightwards double arrow$ $9 straight alpha squared minus 16 straight beta squared equals negative 144 space rightwards double arrow space 16 straight beta squared minus 9 straight alpha squared equals 144 space rightwards double arrow space fraction numerator 16 straight beta squared over denominator 144 end fraction minus fraction numerator 9 straight alpha squared over denominator 144 end fraction equals 1 space rightwards double arrow space straight beta squared over 9 minus straight alpha squared over 16 equals 1$
Hence, the locus of $straight P space left parenthesis straight alpha comma space straight beta right parenthesis$ is $straight y squared over 9 minus straight x squared over 16 equals 1$
which is the equation of a hyperbola in the standard form $space straight y squared over straight a squared minus straight x squared over straight b squared equals 1.$

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Binomial Theorem

Verify that the locus of a point which moves so that the difference of its distances from points A (0, –5) and B (0,5) is always 6, is a hyperbola in the standard form
$space space straight y squared over straight a squared minus straight x squared over straight b squared equals 1.$

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