The number of roots of the equation `1/x+1/(sqrt((1-x^2)))=35/12` is

Options

A. 0
B. 1
C. 2
D. 3

Explanation

Let `a/x=u` and `1/(sqrt((1-x^(2))))=v` then
`u+v=35/12` and `u^(2)+v^(2)=u^(2)v^(2)`
`implies(u+v)^(2)=(35/12)^(2)`
`impliesu^(2)+v^(2)+2uv=(35/12)^(2)`
`impliesu^(2)v^(2)+2uv=(35/12)^(2)[ :' u^(2)+v^(2)=u^(2)v^(2)]`
`impliesu^(2)v^(2)-(35/12)^(2)=0`
`implies(uv+49/12)(uv-25/12)=0`
`impliesuv=-49/12, uv=25/12`
Case I `uv=-49/12,` then
`1/x.1/(sqrt((1-x^(2))))=-49/12` [ here `xlt0`]
`impliesx^(4)-x^(20+((12)^(2))/((49)^(2))=0`
`impliesx=-((5+sqrt(73)))/14
Case II If `uv=25/12` then
`1/x. 1/(sqrt((1-x^(2))))=25/12` [ here `xgt0`]
`impliesx^(4)-x^(2)+((12)^(2))/((25)^(2))=0`
`implies(x^(2)-9/25)(x^(2)-16/25)=0impliesx=3/5,4/5`
On combining both cases
`x=-((5+sqsrt(73)))/14,3/5, 4/5`
Hence number of roots `=3`