Question

The length of the latus rectum of the parabola whose focus is \[\left( {3,3} \right)\] and directrix is \[3x - 4y - 2 = 0\] is:
A) 2
B) 1
C) 4
D) None

Answer 1

Hint: We will first consider the given data that is the focus of the parabola is \[\left( {3,3} \right)\] and the directrix is \[3x - 4y - 2 = 0\] and construct the figure accordingly. We have to find the length of the latus rectum so, we will find the distance between the directrix and the focus point using \[D = |\dfrac{{ax - by - c}}{{\sqrt {{a^2} + {b^2}} }}|\] where \[\left( {x,y} \right) = \left( {3,3} \right)\]. Then we will do the twice of the distance obtained in order to find the latus rectum of the parabola.

Complete step by step solution: We will first consider the given data that is the focus of the parabola is \[\left( {3,3} \right)\] and the directrix is \[3x - 4y - 2 = 0\].
Now, we will construct the figure using the given data of the parabola.

As we have to find the latus rectum of the parabola which is equal to twice the distance between the focus point and directrix of the parabola.
So, we will find the distance between the focus and the directrix of the parabola using the formula, \[D = |\dfrac{{ax - by - c}}{{\sqrt {{a^2} + {b^2}} }}|\].
Thus, we get,
\[
   \Rightarrow D = |\dfrac{{3\left( 3 \right) - 4\left( 3 \right) - 2}}{{\sqrt {{3^2} + {4^2}} }}| \\
   \Rightarrow D = \dfrac{5}{5} \\
   \Rightarrow D = 1 \\
 \]
As the latus rectum of the parabola is equal to the twice of this distance,
Thus, we get,
\[
   \Rightarrow L.R. = 2\left( D \right) \\
   \Rightarrow L.R. = 2\left( 1 \right) \\
   \Rightarrow L.R. = 2 \\
 \]
Hence, we can conclude that the latus rectum is equal to 2.

Thus, option (A) is correct.

Note: Constructing the figure in such questions makes the solution easier. Remember that the relation between the latus rectum, focus and directrix is the latus rectum is equal to twice the distance between the directrix and focus.