Question:

The angle between the lines joining the foci of an ellipse to one particular extremity of the minor axis is $90^{\circ}$ The eccentricity of the ellipse is

Updated On: Sep 3, 2024
  • $\frac{1}{8}$
  • $\frac{1}{\sqrt{3}}$
  • $\sqrt{\frac{2}{3}}$
  • $\sqrt{\frac{1}{2}}$
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The Correct Option is D

Solution and Explanation

In $\Delta OSA$, $tan\, \frac{\pi}{4}=\frac{ae}{b}$
$\Rightarrow 1=\frac{ae}{b}$
$\Rightarrow e^{2}=\frac{b^{2}}{a^{2}}$
$\because e^{2}=1-\frac{b^{2}}{a^{2}}$
$\Rightarrow e^{2}=1-e^{2}$
$e^{2}=\frac{1}{2}$
$\Rightarrow e=\sqrt{\frac{1}{2}}$
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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}