Question

Find the coordinates of the point of intersection of the axis and the directrix of the parabola whose focus is (3, 3) and directrix is 3x − 4y = 2. Find also the length of the latus-rectum. 

Answer 1

The given equation of the directrix is 3x − 4y = 2.  

∴ Slope of the directrix = \[\frac{- 3}{- 4} = \frac{3}{4}\] 

Also, the axis is perpendicular to the directrix.
∴ Slope of the axis = \[\frac{- 4}{3}\] 

The focus lies on the axis of the parabola.
∴ Equation of the axis: \[\left( y - 3 \right) = \frac{- 4}{3}\left( x - 3 \right)\] 

⇒ \[\left( 3y - 9 \right) = - 4x + 12\] 

⇒\[3y + 4x - 21 = 0\] 

Solving equations (1) and (2): 

\[x = \frac{18}{5}, y = \frac{11}{5}\] 

Therefore, the intersection point of the axis and directrix is\[\left( \frac{18}{5}, \frac{11}{5} \right)\].

Also, length of the latus rectum = 2 (Length of the perpendicular from the focus on the directrix) 

= \[2\left| \frac{3\left( 3 \right) + \left( - 4 \right)3 - 2}{\sqrt{16 + 9}} \right| = 2\left| \frac{- 5}{\sqrt{16 + 9}} \right| = 2\]