I am given two points $A(3,5)$ and $H (\sqrt{2},\sqrt{5})$ and a third point $B(h,k)$ . Given that $B$ has rational co-ordinates and A , B are equidistant from H. I need to find $B(h,k)$.
What I tried :
$AH^2 = BH^2$
$h^2 + k^2 - 2\sqrt{2}(h-3) - 2\sqrt{5}(k-5)-34=0$ ------------>(1)
In the book , it is given that since $B(h,k)$ are rational so $h-3=0 , k-5=0$
I didn't understand this step . Eq(1) gives the locus of possible points B. So what does $h-3=0$ and $k-5=0$ got to do with $(h,k)$ being rational?
The sum of a (nonzero) rational number and an irrational number is never zero. Therefore, the expression labeled as equation one is only zero if the irrational terms are each themselves zero. This yields $2\sqrt{2}(h-3)=0$ and $-2\sqrt{5}(k-5)=0$, from which the result follows.