Question

find the equation of ellipse who's foci (+- 3,0) are esentricity 3/5​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that the equation of ellipse be

$\rm :\longmapsto\:\dfrac{ {x}^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} } = 1 \: provided \: that \: a > b$

Now, Given that

$\rm :\longmapsto\:foci \: = \: ( \pm \: 3,0)$

We know,

$\boxed{ \bf{ \: \bf :\longmapsto\:foci \: = \: ( \pm \: ae,0)}}$

So,

$\rm :\longmapsto\:( \pm \: ae,0) \: = \: ( \pm \: 3,0)$

So, On comparing we get,

$\rm :\longmapsto\:ae \: = \: 3 - - - - (1)$

Now, given ghat,

$\rm :\longmapsto\:eccentricity, \: e \: = \: \dfrac{3}{5}$

So, on substituting the value of e in equation (1), we get

$\rm :\longmapsto\:a \times \dfrac{3}{5} = 3$

$\bf\implies \:a \: = \: 5 - - - - (2)$

We know,

$\rm :\longmapsto\: {b}^{2} = {a}^{2}(1 - {e}^{2})$

$\rm :\longmapsto\: {b}^{2} = {a}^{2} - {(ae)}^{2}$

On substituting the values of a and ae, we get

$\rm :\longmapsto\: {b}^{2} = {5}^{2} - {3}^{2}$

$\rm :\longmapsto\: {b}^{2} = 25 - 9$

$\bf :\longmapsto\: {b}^{2} = 16 - - - (3)$

So, on substituting the values in

$\rm :\longmapsto\:\dfrac{ {x}^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} } = 1 \:$

we get

$\bf :\longmapsto\:\dfrac{ {x}^{2} }{ 25} + \dfrac{ {y}^{2} }{ 16 } = 1 \:$

is the required equation of ellipse.

Additional Information :-

For the ellipse,

$\rm :\longmapsto\:\dfrac{ {x}^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} } = 1 \: provided \: that \: a > b$

1. Center = ( 0, 0 )

2. Vertex = ( a, 0 ) and ( - a, 0 ).

3. Length of major axis = 2a

4. Length of minor axis = 2b

5. Length of Latus Rectum = 2b^2 / a

6. Distance between foci = 2ae

7. Distance between directrix = 2a/e.

For the ellipse

$\rm :\longmapsto\:\dfrac{ {x}^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} } = 1 \: provided \: that \: a < b$

1. Center = ( 0, 0 )

2. Vertex = ( 0, b ) and ( 0, - b ).

3. Length of major axis = 2b

4. Length of minor axis = 2a

5. Length of Latus Rectum = 2a^2 / b

6. Distance between foci = 2be

7. Distance between directrix = 2b/e.