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$

(a)\;{(7x + 6y)^2} - 570x + 750y + 2100 = 0 \\

(b)\;{(7x + 6y)^2} + 570x - 750y + 2100 = 0 \\

(c)\;{(7x - 6y)^2} - 570x + 750y + 2100 = 0 \\

(d)\;{(7x - 6y)^2} + 570x - 750y + 2100 = 0 \\

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Hint: For any point on the line of parabola, the distance to the focus is equal to the perpendicular distance to the directrix.

We know that for any point P(x, y) on the line parabola, the distance to the focus is F(3,-4) is equal to the perpendicular distance to the Directrix line d is,

6x - 7y + 5=0

$ \Rightarrow \frac{{{{\left( {6x - 7y + 5} \right)}^2}}}{{\left( {{6^2} + {7^2}} \right)}} = {(x - 3)^2} + {\left( {y + 4} \right)^2}$

Now we know that $\left[ {{{(a + b + c)}^2} = {a^2} + {b^2} + {c^2} + 2(ab + bc + ca)} \right]$ and hence on applying the same formula we have,

$ \Rightarrow 36{x^2} + 49{y^2} + 25 - 84xy - 70y + 60x = 85{x^2} + 85{y^2} - 510x + 2125 + 680y$

And hence on doing the simplification, we have

$ \Rightarrow 49{x^2} + 36{y^2} + 84xy - 570x + 750y + 2100 = 0$

And hence it can be written as,

$ \Rightarrow \;{(7x + 6y)^2} - 570x + 750y + 2100 = 0$

So option a is correct answer.

Note: In this type of question first of all we have to find the directrix as well as the Distance to the focus and with the help of that we can find the equation of Parabola.

We know that for any point P(x, y) on the line parabola, the distance to the focus is F(3,-4) is equal to the perpendicular distance to the Directrix line d is,

6x - 7y + 5=0

$ \Rightarrow \frac{{{{\left( {6x - 7y + 5} \right)}^2}}}{{\left( {{6^2} + {7^2}} \right)}} = {(x - 3)^2} + {\left( {y + 4} \right)^2}$

Now we know that $\left[ {{{(a + b + c)}^2} = {a^2} + {b^2} + {c^2} + 2(ab + bc + ca)} \right]$ and hence on applying the same formula we have,

$ \Rightarrow 36{x^2} + 49{y^2} + 25 - 84xy - 70y + 60x = 85{x^2} + 85{y^2} - 510x + 2125 + 680y$

And hence on doing the simplification, we have

$ \Rightarrow 49{x^2} + 36{y^2} + 84xy - 570x + 750y + 2100 = 0$

And hence it can be written as,

$ \Rightarrow \;{(7x + 6y)^2} - 570x + 750y + 2100 = 0$

So option a is correct answer.

Note: In this type of question first of all we have to find the directrix as well as the Distance to the focus and with the help of that we can find the equation of Parabola.