# What is the locus of the given 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?

• “[the] sum of four times its distance from $x$-axis and nine times its distance from the $y$-axis is equal to 10” —This is an equation written out in words. Just translate it into symbols. Oct 2 '17 at 16:24
• Distance from x-axis is the y-coordinate. So the answer should be..? Oct 2 '17 at 16:25
• im not getting@samjoe how can u write 4y + 9x
– user396850
Oct 2 '17 at 16:27

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

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

• thanks,,,,@HuyQuoc dang
– user396850
Oct 3 '17 at 7:10