Question

If the point $(3,4)$ lies on the locus of the point of intersection of the lines $x\cos \alpha +y\sin \alpha =a$ and $x\sin \alpha -y\cos \alpha =b$, ($\alpha$ is a variable), the point $(a,b)$ lies on the line $3x-4y=0$, then $9a^4+16b^4+34$ is equal to

Answer 1

Point of intersection of the lines $x\cos \alpha +y\sin \alpha =a$ and $x\sin \alpha -y\cos \alpha =b$ is $(a\cos \alpha +b\sin \alpha ,a\sin \alpha -b\cos \alpha )$

$a\cos \alpha +b\sin \alpha =h$ and $a\sin \alpha -b\cos \alpha =k$ (Let)

Square both equation and add,

$\Rightarrow a^2+b^2=h^2+k^2$ ....(Locus)

$(3,4)$ lies on locus,

$a^2+b^2=3^2+4^2=25$

and $(a,b)$ lies on $3x-4y=0\Rightarrow 3a-4b=0$

Solving $a^2+b^2=25$ and $3a-4b=0$

$\Rightarrow a^2=16$ and $b^2=9$

Hence, $9a^4+16b^4+34=3634$