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 \(\dfrac{x^2}{16}+\dfrac{y^2}{9}=1\).
Solution and Explanation
The given equation is \(\dfrac{x^2}{16}+\dfrac{y^2}{9}=1\) or \(\dfrac{x^2}{4^2}+ \dfrac{y^2}{3^2} = 1\)
Here, the denominator of \(\dfrac{x^2}{16}\) is greater than the denominator of \(\dfrac{y^2}{9} .\)
Therefore, the major axis is along the \( x-axis\), while the minor axis is along the \(y-axis.\) On comparing the given equation with
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), we obtain \(a = 4 \) and \(b = 3.\)
\(∴ c = √(a^2 – b^2)\)
\(= √(16-9)\)
\(= √7\)
Therefore,
The coordinates of the foci are\( (√7, 0)\) and \((-√7, 0) \).
The coordinates of the vertices are\( (4, 0)\) and \((-4, 0). \)
Length of major axis =\( 2a= 8 \)
Length of minor axis = \(2b= 6\)
Eccentricity, \(e = \dfrac{c}{a} = \dfrac{√7}{4}\)
Length of the latus rectum \(= \dfrac{2b^2}{a} = \dfrac{(2×3^2)}{4} \)
\( = \dfrac{(2×9)}{4} \)
\(= \dfrac{18}{4} \)
\( = \dfrac{9}{2}\) (Ans)
Top Questions on Ellipse
- Let \( f(x) = x^2 + 9 \), \( g(x) = \frac{x}{x-9} \), and \[ a = f \circ g(10), \, b = g \circ f(3). \]
If \( e \) and \( l \) denote the eccentricity and the length of the latus rectum of the ellipse \[ \frac{x^2}{a} + \frac{y^2}{b} = 1, \] then \( 8e^2 + l^2 \) is equal to: - Let \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a>b \), be an ellipse whose eccentricity is \( \frac{1}{\sqrt{2}} \) and the length of the latus rectum is \( \sqrt{14} \). Then the square of the eccentricity of \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is:
- If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:
- A parabola has vertex \((2, 3)\), equation of directrix is \(2x – y = 1\) and equation of ellipse is \(\frac {x^2}{a^2} +\frac {y^2}{b^2} =1\), \(e=\frac {1}{\sqrt 2}\) and ellipse passing through focur of parabola then square of length of latus rectum of ellipse is
- If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is.
Questions Asked in CBSE Class XI exam
- Draw formulas for the first five members of each homologous series beginning with the following compounds.\((a) H–COOH (b) CH_3COCH_3 (c) H–CH=CH_2\)
- CBSE Class XI
- Organic Chemistry- Some Basic Principles and Techniques
- Distinguish between the following pairs of sentences.
(i) He was visibly moved.
(ii) He was visually impaired.- CBSE Class XI
- The adventure
- How, according to you, can peace and liberty be maintained in a state?
- CBSE Class XI
- The tale of melon City
Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?
- CBSE Class XI
- Pressure
- Describe the change in hybridisation (if any) of the Al atom in the following reaction.
\(AlCl_3+Cl^- \rightarrow AlCl_4^-\)- CBSE Class XI
- Hybridisation
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}