If a normal to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` is `4-3y=7` and its ecentricity is `(sqrt(7))/(4)` , then the volume of L.R can be

Options

A. a) `(9)/(sqrt(2))`
B. b) `(9)/(2)`
C. c) `(8sqrt(337))/(19)`
D. d) `(3sqrt(337))/(8)`

Explanation

Equation of the normal at `(a co theta, b sin theta)` is `ax sec theta-"by cosec" theta=a^(2)-b^(2)`
Comapring with `4x-3y=7` we, get
`(a sec theta)/(4)=(b "cosec" theta)/(3)=(a^(2)-b^(2))/(7)" "(1)`
For`agtb`,
`b^(2)a^(2)(1-e^(2))`
`:. b=(3a)/(4)" "(2)`
From equation (1) we get
`((a)/(4))/(costheta)=((b)/(3))/(sintheta)=sqrt((a^(2))/(16)+(b^(2))/(9))=(a^(2)-b^(2))/(7)" "(3)`
Solging (2) and (3) we get
`rArra=4sqrt(2),b=3sqrt(2)`
L.R. =`(2b^(2))/(a)=(2xx18)/(4sqrt(2))=(9)/(sqrt(2))`
For `altb`, we have,
`a^(2)=b^(2)(1-e^(2))`
`rArra=(3b)/(4)`
`rArrr-7sqrt((9)/(16xx16)b^(2)+(b^(2))/(9))=(9)/(16)b^(2)-b^(2)" "("Using"(3))`
`rArr(sqrt(337))/(6xx16),=a(sqrt(337))/(4)`
`:. b=(sqrt(337))/(3),a=(sqrt(337))/(4)`
`L.R. =(2a^(2))/(b)=(2(337)/(16))/((sqrt(337))/(3))=(sqrt(337))/(8)`