In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $α$ and the number of persons who speak only Hindi is $β$ , then the eccentricity of the ellipse $25 (β^{2} x^{2} + α^{2} y^{2}) = α^{2} β^{2}$ is

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

$\frac{3 \sqrt{15}}{12}$

$\frac{\sqrt{117}}{12}$

$\frac{\sqrt{119}}{12}$

$\frac{\sqrt{129}}{12}$

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

$n (A \cup B) = 100 \begin{array}{l} \Rightarrow 75 + 40 - n (A \cap B) = 100 \Rightarrow n (A \cap B) = 15 \\ α = 60, β = 25 \end{array}$

$25 (β^{2} x^{2} + α^{2} y^{2}) = α^{2} β^{2}$
$\begin{array}{l} 25 [(25)^{2} x^{2} + (60)^{2} y^{2}] = (25)^{2} (60)^{2} \\ [25] (25) [25 x^{2} + 144 y^{2}] = (25)^{2} (60)^{2} \\ \frac{25 x^{2}}{(60)^{2}} + \frac{144 y^{2}}{(60)^{2}} = 1 \frac{x^{2}}{144} + \frac{y^{2}}{25} = 1 \\ e = \sqrt{\frac{144 - 25}{144}} = \frac{\sqrt{119}}{12} \end{array}$