Question:

The length of the semi-latus rectum of an ellipse is one thrid of its major axis, its eccentricity would be

Updated On: Jun 3, 2023

$\frac{2}{3}$
$\sqrt{\frac{2}{3}}$
$\frac{1}{\sqrt{3}}$
$\frac{1}{\sqrt{2}}$

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The Correct Option is A

Solution and Explanation

As per the question

\frac{2 b^{2}}{a}=\frac{1}{3}(2 a)

\frac{b^{2}}{a^{2}}=\frac{1}{3}

Eccentricity is given as

e ^{2} =1-\frac{ b ^{2}}{ a ^{2}}

=1-\frac{1}{3}

=\frac{2}{3}

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Concepts Used:

Ellipse

Ellipse Shape

An ellipse is a locus of a point that moves in such a way that its distance from a fixed point (focus) to its perpendicular distance from a fixed straight line (directrix) is constant. i.e. eccentricity(e) which is less than unity

Properties

Ellipse has two focal points, also called foci.
The fixed distance is called a directrix.
The eccentricity of the ellipse lies between 0 to 1. 0≤e<1
The total sum of each distance from the locus of an ellipse to the two focal points is constant
Ellipse has one major axis and one minor axis and a center

Read More: Conic Section

Eccentricity of the Ellipse

The ratio of distances from the center of the ellipse from either focus to the semi-major axis of the ellipse is defined as the eccentricity of the ellipse.

The eccentricity of ellipse, e = c/a

Where c is the focal length and a is length of the semi-major axis.

Since c ≤ a the eccentricity is always greater than 1 in the case of an ellipse.
Also,
c² = a² – b²
Therefore, eccentricity becomes:
e = √(a² – b²)/a
e = √[(a² – b²)/a²] e = √[1-(b²/a²)]

Area of an ellipse

The area of an ellipse = πab, where a is the semi major axis and b is the semi minor axis.

Position of point related to Ellipse

Let the point p(x₁, y₁) and ellipse

(x² / a²) + (y² / b²) = 1

If [(x₁²/ a²)+ (y₁² / b²) − 1)]

= 0 {on the curve}

<0{inside the curve}

>0 {outside the curve}