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If x cos alpha +y sin alpha = 4 is tange...

If `x cos alpha +y sin alpha = 4` is tangent to `(x^(2))/(25) +(y^(2))/(9) =1`, then the value of `alpha` is

A

`tan^(-1)(3//sqrt(7))`

B

`tan^(-1)(7//3)`

C

`tan^(-1)(sqrt(3)//7)`

D

`tan^(-1)(3//7)`

Text Solution

Verified by Experts

The correct Answer is:
A
`x cos alpha + ysin alpha =4`
`:. y = (-cot alpha) x + 4 cosec alpha`
`:. m =- cot alpha, c = 4 cosec alpha, a^(2) = 25, b^(2) =9`
Now `c^(2) = a^(2)m^(2) + b^(2)`
`:. 16 cosec^(2) alpha = 25 cot^(2) alpha +9`
`:. 16 (1+cot^(2) alpha) = 25 cot^(2) alpha +9`
`:. 7 = 9 cot^(2) alpha rArr cot alpha = (sqrt(7))/(3) rArr tan alpha = (3)/(sqrt(7))`
`:. alpha = tan^(-1) (3//sqrt(7))`
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