Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity, and the length of the latus rectum of the ellipse \(\dfrac{x^2}{4}+\dfrac{y^2}{25}=1\)
Solution and Explanation
The given equation is \(\dfrac{x^2}{4}+\dfrac{y^2}{25}=1\) or \(\dfrac{x^2}{2^2}+\dfrac{y^2}{5^2}=1\)
Here, the denominator of \(\dfrac{y^2}{25}\) is greater than the denominator of \(\dfrac{x}{4}. \)
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis. On comparing the given equation with \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), , we obtain \(b = 2 \) and \(a = 5. \)
\(∴ c = √(a^2 – b^2)\)
\(= √(25-4)\)
\(= √21\)
Therefore, The coordinates of the foci are\( (0, √21)\) and \((0, -√21). \)
The coordinates of the vertices are \((0, 5) \) and \((0, -5)\)
Length of major axis = \(2a= 10 \)
Length of minor axis = \(2b= 4\)
Eccentricity, \(e =\dfrac{c}{1} = \dfrac{√21}{5}\)
Length of the latus rectum\( =\)\( \dfrac{2b^2}{a} = \dfrac{(2×2^2)}{5} \)
\(= \dfrac{(2×4)}{5} = \dfrac{8}{5}\) (Ans. )
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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}