Question

Find the equation of the parabola whose: 

 focus is (2, 3) and the directrix x − 4y + 3 = 0.

Answer 1

 Let P (x, y) be any point on the parabola whose focus is S (2, 3) and the directrix is x − 4y + 3 = 0.
Draw PM perpendicular to x − 4y + 3 = 0.
Then, we have: 

\[SP = PM\]
\[ \Rightarrow S P^2 = P M^2 \]
\[ \Rightarrow \left( x - 2 \right)^2 + \left( y - 3 \right)^2 = \left| \frac{x - 4y + 3}{\sqrt{1 + 16}} \right|^2 \]
\[ \Rightarrow \left( x - 2 \right)^2 + \left( y - 3 \right)^2 = \left( \frac{x - 4y + 3}{\sqrt{17}} \right)^2 \]
\[ \Rightarrow 17\left( x^2 + 4 - 4x + y^2 - 6y + 9 \right) = x^2 + 16 y^2 + 9 - 8xy - 24y + 6x\]
\[ \Rightarrow \left( 17 x^2 - 68x - 102y + 17 y^2 + 13 \times 17 \right) = x^2 + 16 y^2 + 9 - 8xy - 24y + 6x\]
\[ \Rightarrow 16 x^2 + y^2 + 8xy - 74x - 78y + 212 = 0\]