D01-44

Official Solution:


If \(x^2 + y^2 = 100\), all of the following could be true EXCEPT:


A. \(|x| + |y| = 10\)
B. \(|x| \gt |y|\)
C. \(|x| \gt |y| + 10\)
D. \(|x| = |y|\)
E. \(|x| - |y| = 5\)


The equation \(x^2 + y^2 = 100\) represents a circle with a radius of 10 units and centered at the origin. This means that any point (x, y) that satisfies the equation lies on the circle.



Let's analyze each option:

A. \(|x| + |y| = 10\). This is possible if one of the coordinates is 0 and the other is 10 or -10. There are four points on the circle that satisfy this condition: (0, 10), (0, -10), (10, 0), and (-10, 0).

B. \(|x| \gt |y|\). We can see that if \(x = 10\) and \(y = 0\), this option is true. It is also evident from the figure that there must exist points (x, y) on the circle such that \(|x| > |y|\) since the circle cannot consist only of points where \(|x| \leq |y|\).

C. \(|x| \gt |y| + 10\). Since \(|y|\) is more than or equal to 0, then this option implies that \(|x| > 10\), which leads to \(x^2 + y^2\) being greater than 100, contradicting the given equation. Therefore, option C is not possible.

D. \(|x| = |y|\). This option is possible since there must exist (x, y) points on the circle such that \(|x| = |y|\). For example, in the first quadrant, a point at 45 degrees has equal \(x\) and \(y\) coordinates.

E. \(|x| - |y| = 5\). To simplify, let's consider the first quadrant and check whether \(x - y = 5\) is possible. Consider a point (10, 0), which has \(x - y = 10\). As we move this point along the curve in an anticlockwise direction, both \(x\) and \(y\) coordinates change continuously. Specifically, \(x\) decreases from 10 to 0 and \(y\) increases from 0 to 10. Thus, at some point on the curve, there must exist an (x, y) point where \(x - y = 5\).

Therefore, options A, B, D, and E are possible, but option C is not possible.


Answer: C