Question:

The eccentric angle of the point (2,3)(2,\sqrt{3}) lying on x216+y241\frac {x^2}{16}+\frac{y^2}{4}-1 is _________

Updated On: Apr 2, 2024
  • π3\frac {\pi}{3}
  • π6\frac {\pi}{6}
  • π4\frac {\pi}{4}
  • π2\frac {\pi}{2}
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The Correct Option is A

Solution and Explanation

Given equation of an ellipse is
x216+y24=1\frac{x^{2}}{16}+\frac{y^{2}}{4}=1 and point P(2,3)P(2, \sqrt{3})
Let θ\theta be the eccentric angle.
The parametric coordinate of an ellipse is
$\begin{cases}
x=4 \cos \theta \\
y=2 \sin \theta
\end{cases}$...(i)
Given that, eccentric angle at PP is,
2=4cosθcosθ=122=4 \cos \theta \Rightarrow \cos \theta=\frac{1}{2}
2=2sinθsinθ=32\sqrt{2}=2 \sin \theta \Rightarrow \sin \theta=\frac{\sqrt{3}}{2}
Hence, θ=π3\theta=\frac{\pi}{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}