Question

If the distance between the foci of an ellipse is $6$ and the distance between its directrices is $12$ , then the length of its latus rectum is
A. $4\sqrt 3$
B. $\dfrac{4}{{\sqrt 3 }}$
C. $\dfrac{{\sqrt 3 }}{4}$
D. $3\sqrt 2$

Answer 1

Hint : An ellipse is the locus of all those points in a plane such that the sum of their distances from two fixed points in the plane, is constant. The fixed points are known as the foci (singular focus), which are surrounded by the curve. The fixed line is the directrix and the constant ratio is the eccentricity of the ellipse. Eccentricity is a

Complete step-by-step answer :
An ellipse is a two-dimensional shape that is defined along its axes. An ellipse is formed when a cone is intersected by a plane at an angle with respect to its base.
It has two focal points. The sum of the two distances to the focal point, for all the points in the curve, is always constant.
A circle is also an ellipse, where the foci are at the same point, which is the center of the circle.
Ellipse is defined by its two-axis along x and y-axis:
Major axis
Minor Axis
We are given that the distance between the foci of an ellipse $= 6$
Therefore $2ae = 6$
Hence we get $ae = 3$ … ( $1$ )
Also we are given that the distance between its directrices $= 12$
Therefore $\dfrac{{2a}}{e} = 12$
Hence we get $\dfrac{a}{e} = 6$ … ( $2$ )
From the equations ( $1$ )and ( $2$ ) we get
$6e(e) = 3$
Therefore ${e^2} = \dfrac{1}{2}$
And hence from ( $1$ ) we get ${a^2} = 18$
Therefore we have $a = 3\sqrt 2$
Now eccentricity ${e^2} = 1 - \dfrac{{{b^2}}}{{{a^2}}}$
Substituting the value we get ,
$\dfrac{1}{2} = 1 - \dfrac{{{b^2}}}{{18}}$
Therefore we get ${b^2} = 9$
Hence length of latus rectum $= \dfrac{{2{b^2}}}{a}$
$= \dfrac{{2 \times 9}}{{3\sqrt 2 }} = 3\sqrt 2$
Therefore option ( $4$ ) is the correct answer.
So, the correct answer is “Option 4”.

Note : The major axis is the longest diameter of the ellipse (usually denoted by ‘a’), going through the center from one end to the other, at the broad part of the ellipse. Whereas the minor axis is the shortest diameter of ellipse (denoted by ‘b’), crossing through the center at the narrowest part. Half of the major axis is called semi-major axis and half of the minor axis is called the semi-minor axis.