Question

The sum of lengths of major and minor axes- of an ellipse whose eccentricity is $ \frac{4}{5} $ and length of latuserectum is $ 14.4 $ , is

• A. $ 24 $
• B. $ 32 $
• C. $ 64 $
• D. $ 48 $

Answer 1

The Correct Option is C

Given, eccentricity of an ellipse $ e=\frac{4}{5} $
$ \Rightarrow $ $ \frac{c}{a}=\frac{4}{5} $ $ [\because \,\,c=ae] $
$ \Rightarrow $ $ c=\frac{4a}{5} $ ..(i) and length of latuserectum $ \frac{2{{b}^{2}}}{a}=14.4 $
$ \Rightarrow $ $ \frac{{{b}^{2}}}{a}=7.2 $
$ \Rightarrow $ $ \frac{{{b}^{2}}}{a}=\frac{72}{10}=\frac{36}{6} $
$ \Rightarrow $ $ {{b}^{2}}=\frac{36\,a}{5} $ .. (ii) In an ellipse, we know that $ {{c}^{2}}={{a}^{2}}-{{b}^{2}} $
$ \Rightarrow $ $ {{\left( \frac{4a}{5} \right)}^{2}}={{a}^{2}}-\frac{36a}{5} $
$ \Rightarrow $ $ \frac{16{{a}^{2}}}{25}={{a}^{2}}-\frac{36a}{5} $ [using Eqs. (i) and (ii)]
$ \Rightarrow $ $ \frac{16{{a}^{2}}-25{{a}^{2}}}{25}=\frac{-36a}{5} $
$ \Rightarrow $ $ \frac{-9{{a}^{2}}}{25}=-\frac{36\,a}{5} $
$ \Rightarrow $ $ \frac{a}{5}=4 $
$ \Rightarrow $ $ a=20 $ Then, from E (ii), we get $ {{b}^{2}}=\frac{36}{5}\times 20 $
$ \Rightarrow $ $ {{b}^{2}}=144\Rightarrow b=\pm 12 $
$ \Rightarrow $ $ b=12 $ $ [\because \,\,b\ne -12] $ Hence, sum of major and minor axes
$=2(a+b)=2(20+12)=64 $

Answer 2

Ans. A planar curve known as an ellipse surrounds two focus points in such a way that the total of the two distances from any point on the curve to the focal point is a constant. It has the appearance of a circle, a particular kind of ellipse in which both focus points are the same.

• A closed plane curve called an ellipse is created when a point travels while maintaining a constant distance from any two fixed points.
• A closed curve is a plane slice from a right circular cone.
• The curve surrounds the fixed points, which are referred to as "foci."

The equation of ellipse is given by: x²/a² + y²/b² = 1

• The area of an oval-shaped ellipse is determined by the main and minor axes.
• The variable "e" stands for eccentricity, which shows how the ellipse has become longer.
• The conic portion includes the ellipse. For instance, the parabola and the hyperbola are open and unbounded in shape. Typically, an ellipse's equation serves as its definition.

The following two-axis along the x and y-axis define the ellipse.

Major Axis: The longest diameter of the ellipse is known as the major axis. It crosses through the center from one end to another, at the broader part of the ellipse.

Minor Axis: The shortest diameter of the ellipsis is known as the minor axis, which crosses through the center at the narrowest part.

Major Axis: The longest diameter of the ellipse is known as the major axis. It crosses through the center from one end to another, at the broader part of the ellipse.

Minor Axis: The shortest diameter of the ellipsis is known as the minor axis, which crosses through the center at the narrowest part.