Question:
If the normal at one end of a latus-rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through one extremity of the minor axis, then the eccentricity of the ellipse is given by the equation
If the normal at one end of a latus-rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through one extremity of the minor axis, then the eccentricity of the ellipse is given by the equation
Updated On: Apr 8, 2024
- $e^2+e-1=0$
- $e^2+e+1=0$
- $e^4+e^2+1=0$
- $e^4+e^2-1=0$
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Normal at $\left(ae, \frac{b^{2}}{a}\right)$ of ellipse $\frac{x^{2}}{a^{2}} +\frac{y^{2}}{b^{2}} = 1$ is
$\frac{ x-ae}{ae/a^{2}} =\frac{ y-b^{2}/a}{\frac{b^{2}}{a}/b^{2}} \left(\frac{x-x_{1}}{x_{1}/ a^{2}} = \frac{y-y_{1}}{y_{1}/b^{2}}\right) $
It passes thro? $\left(0, -b\right)$ if $\frac{0-ae}{ae/a^{2}} = \frac{-b-b^{2}/a}{1/a}$
$ \Rightarrow -a^{2} = -a\left(b+\frac{b^{2}}{a}\right)$
$ \Rightarrow a= \frac{ab+b^{2}}{a} $
$ \Rightarrow a^{2} = ab+b^{2} = ab +a^{2}-a^{2}e^{2} $
$ \Rightarrow ab = a^{2}e^{2} $
$\Rightarrow b = ae^{2}$
$\Rightarrow b^{2} = a^{2}e^{4} $
$ \Rightarrow a^{2}\left(1-e^{2}\right) = a^{2}e^{4} $
$\Rightarrow 1-e^{2} = e^{4} $
$\Rightarrow e^{4}+e^{2}=1$
$\frac{ x-ae}{ae/a^{2}} =\frac{ y-b^{2}/a}{\frac{b^{2}}{a}/b^{2}} \left(\frac{x-x_{1}}{x_{1}/ a^{2}} = \frac{y-y_{1}}{y_{1}/b^{2}}\right) $
It passes thro? $\left(0, -b\right)$ if $\frac{0-ae}{ae/a^{2}} = \frac{-b-b^{2}/a}{1/a}$
$ \Rightarrow -a^{2} = -a\left(b+\frac{b^{2}}{a}\right)$
$ \Rightarrow a= \frac{ab+b^{2}}{a} $
$ \Rightarrow a^{2} = ab+b^{2} = ab +a^{2}-a^{2}e^{2} $
$ \Rightarrow ab = a^{2}e^{2} $
$\Rightarrow b = ae^{2}$
$\Rightarrow b^{2} = a^{2}e^{4} $
$ \Rightarrow a^{2}\left(1-e^{2}\right) = a^{2}e^{4} $
$\Rightarrow 1-e^{2} = e^{4} $
$\Rightarrow e^{4}+e^{2}=1$
Was this answer helpful?
0
0
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.
View More Questions
Questions Asked in KCET exam
- Ten identical cells each emf 2V and internal resistance 1Ω are connected in series with two cells wrongly connected. A resistor of 10Ω is connected to the combination. What is the current through the resistor?
- KCET - 2023
- Cells in Series and in Parallel
- The speed of sound in an ideal gas at a given temperature T is v. The rms speed of gas molecules at that temperature is vrms. The ratio of the velocities v and vrms for helium and oxygen gases are X and X' respectively. Then \(\frac{X}{X'}\) is equal to
- KCET - 2023
- Thermodynamics
- If \(p(\frac{1}{q}+\frac{1}{r}),q(\frac{1}{r}+\frac{1}{p}),r(\frac{1}{p}+\frac{1}{q})\) are in A.P., then p, q, r
- KCET - 2023
- Arithmetic Progression
- A stretched wire of a material whose Young's modulus Y = 2 × 1011 Nm-2 has Poisson's ratio of 0.25. Its lateral strain εl = 10-3. The elastic energy density of the wire is
- KCET - 2023
- mechanical properties of solids
- A metallic rod of length 1 m held along east-west direction is allowed to fall down freely. Given horizontal component of earth’s magnetic field BH = 3 × 10-5 T. The emf induced in the rod at an instant t = 2s after it is released is ( Take g = 10 ms-2 )
- KCET - 2023
- Faradays laws of induction
View More Questions
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}