Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse \(16x^2 + y^2 = 16\).
Solution and Explanation
The given equation is:
\(16x^2 + y^2 = 16\)
or \(\frac {x^2}{1}+ \frac {y^2}{16} = 1\)
or \(\frac {x^2}{12} + \frac {y^2}{42} = 1\) .......(1)
Here, the denominator of \(\frac {y^2}{4^2}\) is greater than the denominator of \(\frac {x^2}{1^2}\).
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation (1) with \(\frac {x^2}{b^2} + \frac {y^2}{a^2} = 1\), we obtain b = 1 and a = 4.
∴ \(c = \sqrt {(a^2 – b^2)}\)
\(c= \sqrt {(16-1)}\)
\(c= \sqrt {15}\)
Therefore,
The coordinates of the foci are \((0, ±\sqrt {15})\).
The coordinates of the vertices are (0, ±4).
Length of major axis = 2a = 8
Length of the minor axis = 2b = 2
Eccentricity, \(e = \frac ca = \frac {\sqrt {15}}{4}\)
Length of latus rectum = \(\frac {2b^2}{a} = \frac {(2×1^2)}{4} = \frac 24 = \frac 12\)
Top Questions on Ellipse
- Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1 \), \( A < B \), have the same eccentricity \( \sqrt{\frac{1}{3}} \). Let the product of their lengths of latus rectums be \( \sqrt{\frac{32}{3}} \) and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A \), \( B \), \( C \), and \( D \), then the area of the quadrilateral ABCD equals:
- Let the product of the focal distances of the point $$ \left( \sqrt{3}, \frac{1}{2} \right) $$ on the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad (a > b), $$ be $ \frac{7}{4} $. Then the absolute difference of the eccentricities of two such ellipses is:
- Length of latus rectum of ellipse \[ \frac{x^2}{9} + \frac{y^2}{16} = 1 \]
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Questions Asked in CBSE CLASS XII exam
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The act of opening up economies for trading is known as free trade or trade liberalisation. This is done by bringing down trade barriers like tariffs. Trade liberalisation allows goods and services from everywhere to compete with domestic products and services.
Globalisation along with free trade can adversely affect the economies of developing countries by not giving equal playing field by imposing conditions which are unfavourable. With the development of transport and communication systems, goods and services can travel faster and farther than ever before. But free trade should not only let rich countries enter the markets, but allow the developed countries to keep their own markets protected from foreign products.
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Study the given figure carefully and answer the questions that follow :
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Year Number of Towns/UAs Urban Population (in Thousands) % of Total Population Decennial Growth (%) 1931 2,072 33,456.0 11.99 19.12 1941 2,250 44,153.3 13.86 31.97 1951 2,843 62,443.7 17.29 41.42 1961 2,365 78,936.6 17.97 26.41 1971 2,590 1,09,114 19.91 38.23 1981 3,378 1,59,463 23.34 46.14 1991 4,689 2,17,611 25.71 36.47 2001 5,161 2,85,355 27.78 31.13 - CBSE CLASS XII - 2025
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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}