What is the locus of the given question?

Question

I'm attempting to solve the following problem:

What geometric figure is formed by the locus of a point which moves so that the sum of four times its distance from the x-axis and nine times its distance from the y-axis is equal to 10?

My attempt:

From the diagram, I conclude that locus will be a circle.

Is my answer correct or does it contain mistakes? What is the correct solution?

Answer 1

If we call the coordinate of the point is $(x,y)$ then $x$ and $y$ have to satisfy the equation: $4 \mid y \mid +9 \mid x \mid =10$. Now, consider the equation on each quadrant of the $xy$-plane. For instance, on the first quadrant, where $x$ and $y$ are positive, the equation is $4y+9x=10$.

Answer 2

The distance of a point $P(x,y)$ from axis is $|x|$ and $|y|$

Therefore the locus you are looking for has equation $$4|x|+9|y|=10$$ which graphically is a rhombus having as sides the lines

$4x+9y=10;\;4x-9y=10;\;4x+y=-10;\;4x-9y=-10$