Find the co-ordinates of vertices, length of major and minor axes, eccentricity, co-ordinates of foci, equation of directrices and length of latus rectum for each of the following ellipse :
(i) `(x^(2))/(49)+y^(2)/(25)=1`
(ii) `(x^(2))/(4)+(y^(2))/(9)=1`

Explanation

(i) Equation of ellipse
`(x^(2))/(49)+y^(2)/(25)=1`
(ii) `rArr" "(x^(2))/(7^(2))+(y^(2))/(5^(2))=1`
Here a=7, b=5
`:." "agtb`.
Co-ordinate of vertices `(pma,0)=(pm7,0)`.
Major axis `=2a=2xx7=14`.
Minor axis `=2b=2xx5=10`.
Eccentricity `e=sqrt(1-(b^(2))/(a^(2)))=sqrt(1-(25)/(49))`
`=sqrt((24)/(49))=(2sqrt(6))/(7)`.
Foci `=(pmae,0)`
`=(pm7*(2sqrt(6))/(7),0)=(pm2sqrt(6),0)`.
Equation of directrices `x=pm(a)/(e)`
`rArr" "x=pm((7)/(2sqrt(6)))/(7)`
`rArr" " rArr" "x=pm(49)/(2sqrt(6))`.
Length of latus rectum `=(2b^(2))/(a)=(2xx25)/(7)=(50)/(7)`.
(ii) Equation of ellipse
`(x^(2))/(4)+y^(2)/(9)=1`
`rArr" "(x^(2))/(2^(2))+(y^(2))/(3^(2))=1`
Here a=2, b=3
`:." "altb`
Co-ordinates of vertices `=(0,pmb)=(0,pm3)`.
Major aixs `=2b=2xx3=6`.
Minor axis `=2a=2xx2=4`.
Eccentricity e `=sqrt(1-(a^(2))/(b^(2)))=sqrt(1-(4)/(9))`
`=sqrt((5)/(9))=(sqrt(5))/(3)`.
Co-ordinates of foci
`=(0,pmbe)=(0,pm3.(sqrt(5))/(3))=(0pmsqrt(5))`.
Equation of directrices `y=pm(b)/(e)`
`rArr" "y=pm(3)/(sqrt(5)/(3))`
`rArr" "y=pm(9)/(sqrt(5))`.
Length of latus rectum `=(2a^(2))/(b)=(2xx4)/(3)=(8)/(3)`.