Question

23. The latus rectum subtends a right angle
at the centre of the ellipse then its
eccentricity is
1) 2 sin 18
2) 2 cos 18
3) 2 sin 54
4) 2 cos 54°​

Answer 1

$\sf\red{\underbrace{Answer \implies 2\:sin\:18\degree}}$

$\sf{\underline{\underline{Given:}}}$

$\sf{The\:latus\:rectum\:subtends\:a\:right\:angle}$$\sf{at\:the\:centre\:of\:the\:ellipse}$

$\sf{\underline{\underline{To\:Find:}}}$

$\sf{Eccentricity = \:......?}$

$\sf{\underline{\underline{SoLUtION:}}}$

$\sf{ If\:latus\:rectum\:subtends\:right\:angle}$$\sf{at\:the\:centre,\:then,}$

$\sf\purple{\implies tan\:45\degree = \dfrac{\dfrac{{b}^{2}}{{a}^{2}}}{ae}}$

$\sf\orange{\implies e = \dfrac{{b}^{2}}{{a}^{2}} = 1 - {e}^{2}}$

$\sf\green{\implies {e}^{2} + e - 1 = 0}$

$\sf\red{\implies e = \dfrac{ -1 + \sqrt{5}}{2} = 2\:sin\:18\degree}$

$\sf{\underline{\underline{Hence:}}}$

$\sf\green{Eccentricity = 2\:sin\:18\degree}$