Four times the distance from the $x$-axis plus 9 times the distance from $y$-axis equals $10$. What geometric figure is formed by the locus of points such that the sum of four times the distance from the $x$-axis and nine times its distance from $y$-axis is equal to $10$?    

I get $4x+9y=10$. So it is a straight line, but the given answer is parallelogram. Can anyone tell me where my mistake is?

If $P$ has coordinates $(x,y)$, then $d(P,x\text{ axis})=|y|$ and $d(P,y\text{ axis})=|x|$. So your stated condition requires $4|y|+9|x|=10$, the graph of which is shown below.

