Binomial Theorem

Question
CBSEENMA11015120

Find the equation of the parabola whose focus is at point S (2, 5) and the directrix is line 3x + 4y + 1 = 0.

Solution

Let line l be the directrix with equation
3x + 4y + 1 = 0 and S (2, 5) be the focus.
Take a point straight P space left parenthesis straight alpha comma space straight beta right parenthesis on the parabola
∴ By definition, PS = 'PM   rightwards double arrow d = p
rightwards double arrow             square root of left parenthesis straight alpha minus 2 right parenthesis squared plus left parenthesis straight beta minus 5 right parenthesis squared end root space equals space open vertical bar fraction numerator 3 straight alpha plus 4 straight beta plus 1 over denominator square root of left parenthesis 3 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root end fraction close vertical bar
rightwards double arrow               left parenthesis straight alpha minus 2 right parenthesis squared plus left parenthesis straight beta minus 5 right parenthesis squared space equals space fraction numerator left parenthesis 3 straight alpha plus 4 straight beta plus 1 right parenthesis squared over denominator 25 end fraction
rightwards double arrow          25 straight alpha squared plus 25 straight beta squared minus 100 straight alpha minus 250 straight beta plus 725 equals 9 straight alpha squared plus 16 straight beta squared plus 24 αβ plus 6 straight alpha plus 8 straight beta plus 1
rightwards double arrow              space space space space space space 16 straight alpha squared plus 9 straight beta squared minus 24 αβ minus 106 straight alpha minus 258 straight beta plus 724 equals 0
Hence, the locus of P or the equation of parabola is :
16 straight x squared plus 9 straight y squared minus 24 xy minus 106 straight x minus 258 straight y plus 724 equals 0
                                          

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