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(i) Let P (x, y) be any point on the parabola whose focus is S (3, 0) and the directrix is 3x + 4y = 1.
Draw PM perpendicular to 3x + 4y = 1.
Then, we have:
$SP=PM\phantom{\rule{0ex}{0ex}}⇒S{P}^{2}=P{M}^{2}\phantom{\rule{0ex}{0ex}}⇒{\left(x-3\right)}^{2}+{\left(y-0\right)}^{2}={\left(\frac{3x+4y-1}{\sqrt{9+16}}\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒{\left(x-3\right)}^{2}+{y}^{2}={\left(\frac{3x+4y-1}{5}\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒25\left\{{\left(x-3\right)}^{2}+{y}^{2}\right\}={\left(3x+4y-1\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒\left(25{x}^{2}-150x+25{y}^{2}+225\right)=9{x}^{2}+16{y}^{2}+1+24xy-8y-6x\phantom{\rule{0ex}{0ex}}⇒16{x}^{2}+9{y}^{2}-24xy-144x+8y+224=0$

(ii) Let P (x, y) be any point on the parabola whose focus is S (1, 1) and the directrix is xy + 1 = 0.
Draw PM …

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