Question

In an ellipse, the distance between the foci is 8 and the distance between the directrices is 25. then the length of major axis is:

Answer 1

Let equation of ellipse is x²/a² + y²/b² = 1
Then, co-ordinate of two foci are (ae , 0) ,(-ae,0) [ e is ecentricty of ellipse ]
so, distance between two foci = ae - (-ae) = 2ae

And equation of directrices are x = a/e , -a/e
Distance between directrices = 2a/e
Length of major axis = 2a

Now, come to the point
Distance between foci = 2ae = 8 ----(1)
distance between directrices = 2a/e = 25 -----(2)

Multiply equations (1) and (2)
4a² = 8 × 25
⇒a² = 50
⇒ a = ±5√2

So, length of major axis = |2a| [ ∵ length is always positive]
= 2 × 5√2 = 10√2 unit

Hence, answer is 10√2

Answer 2

Answer:

$The \: general \: equation \: of \: an \: \\ ellipse \: is$

$\frac{x {}^{2} }{a {}^{2} } + \frac{x {}^{2} }{b {}^{2} } = 1$

$If the major axis is along the x-axis then b2=a2(1−e2)b2=a2(1−e2), the foci are (±ae,0)(±ae,0) and directrices are x= \frac{a}{e}$

$If the major axis is along the y-axis then a2=b2(1−e2)a2=b2(1−e2), the foci are (0,±be)(0,±be) and directrices are y= \frac{b}{e}$