If $x \cos A-y \sin A=1$ and $x \sin A+y \cos A=4$, then the value of $17 x^2+17 y^2$ is:

289

We are given that ,

x cosA - y sinA = 1

& x sinA + y cosA = 4

Square both the equations and then add,

x²cos²A + y²sin²A - 2xcosA.ysinA + x²sin²A + y²cos²A + 2xcosA.ysinA 

= 1 + 16

x² (sin²A +  cos²A )   + y² ( sin²A +  cos²A ) = 17

{ Using  , sin²A +  cos²A = 1 }

x² + y² = 17

Now,

17 (x² + y² )

= 17 × 17

= 289