In a triangle `ABC, (a)/(b) = (2)/(3) and sec^(2) A = (8)/(5)`. Find the number of triangle satisfying these conditions

Answer

two

Explanation

We have `(a)/(b) = (b)/(3) = k` and
`sec^(2) A = (8)/(5)`
`rArr cos^(2) A = (5)/(8)`
`rArr (5)/(8) = ((9k^(2) + c^(2) - 4k^(2))/(6kc))^(2) = ((5k^(3) + c^(2))/(6kc))^(2)`
`rArr 45k^(2) c^(2) = 50 k^(4) + 20 k^(2) c^(2) + 2c^(4)`
`rArr 2c^(4) - 25 k^(2) c^(2) + 50k^(4) = 0`
`rArr c^(2) = (25 k^(2) +- sqrt(625 k^(4) - 400 k^(4)))/(4)`
`= (25k^(2) +- 15 k^(2))/(4) = 10 k^(2), (5)/(2) k^(2)`
There are two possible valid values of `c^(2)`. Hence there exist two triange satisfying the given conditions