Let $S$ and $S'$ be the foci of an ellipse and B be one end of its minor axis. If $SBS'$ is an isosceles right angled triangle then the eccentricity of the ellipse is
- $\frac{1}{\sqrt{2}}$
- $\frac{1}{2}$
- $\frac{\sqrt{3}}{2}$
- $\frac{1}{3}$
The Correct Option is A
Solution and Explanation
$S B S'$ is an isosceles right angle triangle.
$\therefore S S^{2}=S B^{2}+S^{\prime} B^{2}$
$\Rightarrow (2 \,a e)^{2}=b^{2}+\left(a e^{2}+b^{2}+(a e)^{2}\right.$
$\Rightarrow 4(a e)^{2}=2\left(b^{2}+(a e)^{2}\right)$
$\Rightarrow (a e)^{2}=b^{2}$
$\Rightarrow e^{2}=\frac{b^{2}}{a^{2}}$
$\Rightarrow 1-e^{2}=1-\frac{b^{2}}{a^{2}}$
$\Rightarrow 1-e^{2}=e^{2} \left[\because e=\sqrt{1-\frac{b^{2}}{a^{2}}}\right]$
$\Rightarrow 2e^{2}=1$
$\Rightarrow e = \frac{1}{\sqrt{2}}$
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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}