I'm trying to figure out this question: in the Cartesian coordinate system, how many grid points (x,y) satisfy $(|x|-1)^2 + (|y|-1)^2 < 2$? When I plugged it in to a graphing calculator, it didn't let me use the < sign, so I changed it to the = sign, and got a graph that looked like four connected semicircles, with centers at (1,1), (1,-1), (-1,1), and (-1,-1). The resulting image looked like a four-leafed clover.
Asked
Active
Viewed 233 times
2
-
I don't know how to insert a screenshot image, but you can put the equation (with an = sign) into https://www.desmos.com/calculator. – Jason Chen May 06 '14 at 04:31
-
By grid points do you mean lattice points, as in, both $x, y$ are integers? – Christopher Liu May 06 '14 at 04:32
-
yes, x and y both need to be integers. – Jason Chen May 06 '14 at 04:32
-
Take the resulting image and count the grid points that lie inside the shape, but not on the outline. – Graham Kemp May 06 '14 at 05:01
2 Answers
2
Hint: Notice that if $(x,y)$ is on the boundary of the region, then so are the points $(\pm x,\pm y)$. This means that the region is symmetric with respect to the $x$ and $y$ axes. Hence, you may graph it in the first quadrant (where $x=|x|$ and $y=|y|$), and reflect to the other three quadrants to obtain the whole shape.
From a quick visualization, I believe that the answer is $16$. You should check this on your own to make sure it is right.
Jared
- 31,451
-
@GrahamKemp: $(0,0)$ is not a solution. I made the same mistake in my first count as well. – Jared May 06 '14 at 05:45
-
Ack! Of course. The overlapping circles all intersect right at the origin. Plotting $(|x|-1)^2+(|y|-1)^2 < 1.5$ makes it clear that there's an interior non-overlapping area that, in the case of $<2$, is just the origin. – Graham Kemp May 06 '14 at 21:23
0
Hint: Notice the powers of 2. What does squaring a number do? Try listing out all possible values of $y$ for $x = -2, -1, \ldots, 2$.
Christopher Liu
- 1,319