Question

Find the equation of the parabola whose focus is the point (2, 3) and directrix is the line x − 4y + 3 = 0. Also, find the length of its latus-rectum.

 

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: 

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

Length of the latus rectum = 2(Length of the perpendicular from the focus on the directrix)
                                           = 2(Length of the perpendicular from (2, 3) on the directrix)
                                           =\[2\left| \frac{2 - 12 + 3}{\sqrt{16 + 1}} \right| = 2\left| \frac{- 7}{\sqrt{17}} \right| = 2\left( \frac{7}{\sqrt{17}} \right) = \frac{14}{\sqrt{17}}\]