Question

Find the coordinates of the centre, vertices, eccentricity, foci, length of the latus rectum of the ellipse 25x2 +16y2 =1600.

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Centre=(0,0)}}}$

$\green{\tt{\therefore{Vertices=(0,\pm10)}}}$

$\green{\tt{\therefore{Foci=(0,\pm2\sqrt{5})}}}$

$\green{\tt{\therefore{Latus\:rectum(LL')=12.8\:units}}}$

$\green{\tt{\therefore{Eccentricity=\frac{1}{\sqrt{5}}}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green {\underline \bold{Given : }} \\ \tt{ : \implies Eqn \: of \: ellipse = 25{x}^{2} + 16{y}^{2} = 1600} \\ \\ \red {\underline \bold{to \: find: }}\\ \tt{:\implies Centre=?} \\\\ \tt{:\implies Vertices=?}\\ \\ \tt{:\implies Foci=?}\\ \\ \tt {: \implies Length \: of \: latus \: rectum (LL')=?}\\\\ \tt{:\implies Eccentricity=?}$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt{: \implies Centre = (0,0)} \\ \\ \green{\tt{: \implies Centre = (0,0)}}$

$\tt{:\implies 25x^{2}+16y^{2}=1600}\\\\ \tt{:\implies \frac{x^{2}}{\frac{1600}{25}}+\frac{y^{2}}{\frac{1600}{16}}=1}\\\\ \tt{: \implies \frac{ {x}^{2} }{64} + \frac{ {y}^{2} }{100} = 1} \\ \\ \text{So, \: it \: is \: in \: the \: form \: of} \\ \tt{\to \frac{ {x}^{2} }{ {a}^{2} } + \frac{ {y}^{2} }{ {b}^{2} } = 1} \\ \\ \bold{Where : } \\ \tt{\circ \: {a}^{2} = 64} \\ \\ \tt{\circ \: {b}^{2} = 100} \\\\ \tt{\circ\: a< b}$

$\bold{As \: we \: know \: that} \\ \tt{: \implies vertices = (0, \pm b)} \\ \\ \green{\tt{: \implies vertices = (0, \pm 10)}} \\ \\ \bold{As \: we \: know \: that} \\ \tt{ : \implies {a}^{2} = {b}^{2}(1 - {e}^{2} )} \\ \\ \tt{: \implies64 = 100(1 - {e}^{2} )} \\ \\ \tt{: \implies {e}^{2} = 1 - \frac{4}{5} } \\ \\ \tt{: \implies {e}^{2} = \frac{1}{5} } \\ \\ \green{\tt{: \implies e = \frac{1}{\sqrt{5}} }}\\ \\ \bold{As \: we \: know \: that} \\ \tt{: \implies foci = (0, \pm be)} \\ \\ \green{\tt{: \implies Foci= (0, \pm 2 \sqrt{5} )}}$

$\bold{As \: we \: know \: that} \\ \tt{ : \implies Latus \: rectum = \frac{2 {a}^{2} }{b} } \\ \\ \text{Putting \: given \: values} \\ \tt{ : \implies Latus \: rectum = \frac{2 \times 64}{10} } \\ \\ \green{\tt{ : \implies Latus \: rectum = 12.8\:units}}$