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Question
Find the equation of the parabola whose:
focus is (3, 0) and the directrix is 3x + 4y = 1
Solution
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:
\[SP = PM\]
\[ \Rightarrow S P^2 = P M^2 \]
\[ \Rightarrow \left( x - 3 \right)^2 + \left( y - 0 \right)^2 = \left( \frac{3x + 4y - 1}{\sqrt{9 + 16}} \right)^2 \]
\[ \Rightarrow \left( x - 3 \right)^2 + y^2 = \left( \frac{3x + 4y - 1}{5} \right)^2 \]
\[ \Rightarrow 25\left\{ \left( x - 3 \right)^2 + y^2 \right\} = \left( 3x + 4y - 1 \right)^2 \]
\[ \Rightarrow \left( 25 x^2 - 150x + 25 y^2 + 225 \right) = 9 x^2 + 16 y^2 + 1 + 24xy - 8y - 6x\]
\[ \Rightarrow 16 x^2 + 9 y^2 - 24xy - 144x + 8y + 224 = 0\]
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