Let each of the two ellipses and have eccentricity . Let the lengths of the latus recta of and be and , respectively, such that If the distance between the foci of is 8, then the distance between the foci of is [2026]

Question

Let each of the two ellipses $E_{1} : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, (a > b)$ and $E_{2} : \frac{x^{2}}{A^{2}} + \frac{y^{2}}{B^{2}} = 1, (A < B)$ have eccentricity $\frac{4}{5}$ . Let the lengths of the latus recta of $E_{1}$ and $E_{2}$ be $l_{1}$ and $l_{2}$ , respectively, such that $2 l_{1}^{2} = 9 l_{2} .$ If the distance between the foci of $E_{1}$ is 8, then the distance between the foci of $E_{2}$ is [2026]

Smart Achievers · Accepted Answer

(4)

$2 a e = 8 \Rightarrow a = 5$

$b^{2} = a^{2} (1 - e^{2})$

$b^{2} = a^{2} \times \frac{9}{25} \Rightarrow b^{2} = 9$

$E_{1} : \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$

$ℓ_{1} : \frac{2 b^{2}}{a} = \frac{2 \times 9}{5} = \frac{18}{5}$

$A^{2} = B^{2} (1 - e^{2}) \Rightarrow A^{2} = \frac{9}{25} B^{2} \Rightarrow A = \frac{3}{5} B$

$2 ℓ^{2} = 9 ℓ_{2} \Rightarrow 2 {(\frac{18}{5})}^{2} = 9 ℓ_{2} \Rightarrow ℓ_{2} = \frac{4 \times 18}{25}$

$\frac{2 A^{2}}{B} = \frac{72}{25} \Rightarrow A^{2} = \frac{36}{25} B$

$\frac{9}{25} B^{2} = \frac{36}{25} B \Rightarrow B = 4$

$Distance between foci = 2 B e = 2 \times \frac{4}{5} \times 4 = \frac{32}{5}$

Solution

1 $\frac{8}{5}$

2 $\frac{96}{5}$

3 $\frac{16}{5}$

4 $\frac{32}{5}$

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Solution

Q. Let each of the two ellipses E1:x2a2+y2b2=1, (a>b) and E2:x2A2+y2B2=1, (A<B) have eccentricity 45. Let the lengths of the latus recta of E1 and E2 be l1 and l2, respectively, such that 2l1 2=9l2. If the distance between the foci of E1 is 8, then the distance between the foci of E2 is [2026]

1 85

2 965

3 165

4 325

Ans. (4) 2ae=8⇒a=5 b2=a2(1-e2) b2=a2×925⇒b2=9 E1: x225+y29=1 ℓ1: 2b2a=2×95=185 A2=B2(1-e2)⇒A2=925B2⇒A=35B 2ℓ2=9ℓ2⇒2(185)2=9ℓ2 ⇒ℓ2=4×1825 2A2B=7225⇒A2=3625B 925B2=3625B⇒B=4 Distance between foci =2Be=2×45×4=325

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1 $\frac{8}{5}$

2 $\frac{96}{5}$

3 $\frac{16}{5}$

4 $\frac{32}{5}$