Question

The length of the latus rectum of the parabola, whose focus is \[\left( {\dfrac{{{u^2}}}{{2g}}sin2\alpha , - \dfrac{{2{u^2}}}{g}co{s^2}\alpha } \right)\] and directrix is\[\;y = \dfrac{{{u^2}}}{{2g}}\], is
A.\[\dfrac{{{u^2}}}{g}co{s^2}\alpha \]
B.\[\dfrac{{{u^2}}}{g}cos2\alpha \]
C.\[\dfrac{{2{u^2}}}{g}cos2\alpha \]
D.\[\dfrac{{2{u^2}}}{g}co{s^2}\alpha \]

Answer 1

Hint: To solve this question or to get the length of latus rectum we have to use the relation that length of latus rectum in a parabola is equal to 2 times the perpendicular distance between the focus and the directrix of the parabola. To get the perpendicular length between the directrix and the focus of the parabola use the formula which we are using to find the perpendicular length from a point to the given line.

Complete step-by-step answer:
First we need to find the distance between the focus and the directrix line or we can say the perpendicular distance from the focus to the directrix line.
Here the focus is given \[F = \left( {\dfrac{{{u^2}}}{{2g}}sin2\alpha , - \dfrac{{{u^2}}}{{2g}}cos2\alpha } \right)\]
And the directrix is given as \[\;y = \dfrac{{{u^2}}}{{2g}}\]
We know that the perpendicular distance from the point \[\left( {{x_1},{y_1}} \right)\] on the line \[a{x_1} + b{y_1} + c = 0\] is
\[\Rightarrow \dfrac{{\left| {a{x_1} + b{y_1} + c = 0} \right|}}{{\sqrt {{a^2} + {b^2}} }}\]
So we will find the perpendicular distance from the focus to the directrix line
First we convert the directrix line in the standard form of a line
\[\Rightarrow \;x \times 0 + y - \dfrac{{{u^2}}}{{2g}} = 0\]
So the perpendicular distance will be
\[
   = \dfrac{{\left| {\dfrac{{{u^2}}}{{2g}}sin2\alpha \times 0 - \dfrac{{{u^2}}}{{2g}}cos2\alpha - \dfrac{{{u^2}}}{{2g}}} \right|}}{{\sqrt {{0^2} + {1^2}} }} \;
  \]
On simplification we get,
\[ = \dfrac{{{u^2}}}{{2g}}\left( {cos2\alpha + 1} \right)\]
Now we have to calculate the length of latus rectum
We know that the length of laltus racum =2( perpendicular distance from the focus to the directrix)
So the length of latus rectum \[ = 2 \times \left[ {\dfrac{{{u^2}}}{{2g}}\left( {cos2\alpha + 1} \right)} \right]\]
Here 2 will get cancelled out
Then on simplification
We get,
Length of latus rectum = \[\dfrac{{2{u^2}}}{g}co{s^2}\alpha \] [As we know that \[cos2\alpha + 1 = 2co{s^2}\alpha \]]
So, the correct answer is “Option D”.

Note: It is a property of a parabola that the length of the latus rectum of a parabola is always two times the perpendicular length of from focus to the directrix of the parabola. This needs to know the students to solve this type of question.