Answer
Verified
417.3k+ views
Hint – We know the Focus = (ae, 0) and vertex = (a, 0) and the distance between focus and vertex = a(1-e) and eccentricity of the ellipse ${e^2} = 1 - \dfrac{{{b^2}}}{{{a^2}}}$. Use this to find the answer.
Complete step-by-step answer:
Now, according to the question,
Length of the latus rectum of an ellipse = 4 units.
Distance between a focus and its nearest vertex on the major axis is 3/2 units.
For better understanding refer the figure below-
Now we know that the Focus = (ae, 0) and vertex = (a, 0) and the distance between focus and vertex = a(1-e).
Now,
$
a(1 - e) = \dfrac{3}{2} \\
\Rightarrow a - ae = \dfrac{3}{2} \\
\Rightarrow a - \dfrac{3}{2} = ae \to (1) \\
$
Squaring the above equation, we get-
$ \Rightarrow {a^2} + \dfrac{9}{4} - 3a = {a^2}{e^2} \to (2)$
Now, using the length of the latus rectum is given by $\dfrac{{2{b^2}}}{a}$.
We know the length of the latus rectum as given in the question is 4 units.
Therefore, $
\dfrac{{2{b^2}}}{a} = 4 \\
\Rightarrow {b^2} = 2a \to (3) \\
$
Now, we know eccentricity $
{e^2} = 1 - \dfrac{{{b^2}}}{{{a^2}}} \\
= 1 - \dfrac{{2a}}{{{a^2}}} \\
${from equation (3)}
$ \Rightarrow {e^2} = 1 - \dfrac{2}{a} \to (4)$
Substituting this in equation (2), we get-
$
\Rightarrow {a^2} + \dfrac{9}{4} - 3a = {a^2}\left( {1 - \dfrac{2}{a}} \right) \\
\Rightarrow {a^2} + \dfrac{9}{4} - 3a = {a^2} - 2a \\
\Rightarrow a = \dfrac{9}{4} \\
$
Therefore,
$
\Rightarrow {e^2} = 1 - \dfrac{2}{a} \\
\Rightarrow {e^2} = 1 - \dfrac{2}{{\dfrac{9}{4}}} \\
\Rightarrow {e^2} = 1 - \dfrac{2}{9} \times 4 \\
\Rightarrow {e^2} = 1 - \dfrac{8}{9} = \dfrac{1}{9} \\
\Rightarrow e = \dfrac{1}{3} \\
$
Hence, the eccentricity of the given ellipse is $e = \dfrac{1}{3}$.
Note – Whenever such types of questions appear, then write the given things in the question and then by using the standard formula of the distance between focus and vertex = a(1-e) = 3/2. Squaring both the sides, we get, ${a^2} + \dfrac{9}{4} - 3a = {a^2}{e^2}$ and then using the standard formula of eccentricity of the ellipse ${e^2} = 1 - \dfrac{{{b^2}}}{{{a^2}}}$, find the value of e.
Complete step-by-step answer:
Now, according to the question,
Length of the latus rectum of an ellipse = 4 units.
Distance between a focus and its nearest vertex on the major axis is 3/2 units.
For better understanding refer the figure below-
Now we know that the Focus = (ae, 0) and vertex = (a, 0) and the distance between focus and vertex = a(1-e).
Now,
$
a(1 - e) = \dfrac{3}{2} \\
\Rightarrow a - ae = \dfrac{3}{2} \\
\Rightarrow a - \dfrac{3}{2} = ae \to (1) \\
$
Squaring the above equation, we get-
$ \Rightarrow {a^2} + \dfrac{9}{4} - 3a = {a^2}{e^2} \to (2)$
Now, using the length of the latus rectum is given by $\dfrac{{2{b^2}}}{a}$.
We know the length of the latus rectum as given in the question is 4 units.
Therefore, $
\dfrac{{2{b^2}}}{a} = 4 \\
\Rightarrow {b^2} = 2a \to (3) \\
$
Now, we know eccentricity $
{e^2} = 1 - \dfrac{{{b^2}}}{{{a^2}}} \\
= 1 - \dfrac{{2a}}{{{a^2}}} \\
${from equation (3)}
$ \Rightarrow {e^2} = 1 - \dfrac{2}{a} \to (4)$
Substituting this in equation (2), we get-
$
\Rightarrow {a^2} + \dfrac{9}{4} - 3a = {a^2}\left( {1 - \dfrac{2}{a}} \right) \\
\Rightarrow {a^2} + \dfrac{9}{4} - 3a = {a^2} - 2a \\
\Rightarrow a = \dfrac{9}{4} \\
$
Therefore,
$
\Rightarrow {e^2} = 1 - \dfrac{2}{a} \\
\Rightarrow {e^2} = 1 - \dfrac{2}{{\dfrac{9}{4}}} \\
\Rightarrow {e^2} = 1 - \dfrac{2}{9} \times 4 \\
\Rightarrow {e^2} = 1 - \dfrac{8}{9} = \dfrac{1}{9} \\
\Rightarrow e = \dfrac{1}{3} \\
$
Hence, the eccentricity of the given ellipse is $e = \dfrac{1}{3}$.
Note – Whenever such types of questions appear, then write the given things in the question and then by using the standard formula of the distance between focus and vertex = a(1-e) = 3/2. Squaring both the sides, we get, ${a^2} + \dfrac{9}{4} - 3a = {a^2}{e^2}$ and then using the standard formula of eccentricity of the ellipse ${e^2} = 1 - \dfrac{{{b^2}}}{{{a^2}}}$, find the value of e.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Two charges are placed at a certain distance apart class 12 physics CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is BLO What is the full form of BLO class 8 social science CBSE