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  • In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at <img alt="\mathrm{(0,5 \sqrt{3})}" src="https://entrancecorne

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at \mathrm{(0,5 \sqrt{3})}, then the length of its latus rectum is

Option: 1

3


Option: 2

5


Option: 3

4


Option: 4

6


Answers (1)

\mathrm{\text { Let equation of ellipse is } \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \quad(a>b)}

Now, \mathrm{2 a-2 b=10 \Rightarrow a-b=5}              ....(i)

Also, \mathrm{a e=5 \sqrt{3}}                                                  ....(ii)

Now, \mathrm{a^2 e^2=a^2-b^2 \Rightarrow(5 \sqrt{3})^2=a^2-b^2}(Using (ii))

\mathrm{\Rightarrow \quad(a+b)(a-b)=75 \Rightarrow a+b=15}.(iii) (Using (i))

Solving (i) and (iii), we get \mathrm{a=10, b=5}

Length of latus rectum

\mathrm{ =\frac{2 b^2}{a}=\frac{2 \times(5)^2}{10}=\frac{50}{10}=5 }

Posted by

Ramraj Saini

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