Probability

Hi,
I can't draw a figure and upload it but I will try to explain it. I will be easy to visualize if we can a draw a figure.
x^2 + y^2 = 4 is circle with center at origin and radius '2' units. It intersects X-axis at (-2,0) & (2,0) as well as Y-axis at (0,2) and (0.-2)
Consider the lines y=x+2. It passes through (-2,0) and (0,2). the region above this line is y>x+2
Consider point (m,n) on the circle.
n>m+2 is the set of points on the circle, over the line y = x+2.
It is one quarter of the circumference of the circle.
Si, probability than n>m+2 is 1/4.

is it a gmat type question?

Will respectfully disagree with Brent here - while I can't remember any such questions on the actual exam, I know I've seen practice questions (from other reputable GMAT prep companies, not just ours) that involve the equation of a circle centered on the origin ... so SOMEBODY must've seen them (at least, you'd suppose).

Anyway, let's do this algebraically.

If m² + n² = 4, and n > m + 2, then

n² = 4 - m²
n = sqrt(4-m²)

Let's substitute this in our equality

sqrt(4 - m²) > m + 2
4 - m² > m² + 4m + 4
0 > 2m² + 4m
0 > 2m(m + 2)
0 > m(m+2)

So m is negative and (m+2) is positive, in which case

0 > m and
m + 2 > 0, or m > -2, so
0 > m > -2

So this works for 0 > m > -2. If 0 > m > -2 and n > m + 2, then 2 > n > 0.

0 > m > -2 and 2 > n > 0 accounts for 1/4 of the circumference of the circle (the portion in Quadrant II), so that's our answer.

Great question!

This problem seems beyond the scope of the GMAT, but there is still a good take-away that can be gleaned here:
Many geometry problems are best solved by DRAWING.

x² + y² = r² is the equation of a circle centered at the origin with a radius of length r.
Thus, x² + y² = 4 is the equation of a circle centered at the origin with a radius of 2.

The problem asks for the probability that a point on this circle will satisfy the constraint that y > x + 2.
y > x + 2 is the region ABOVE the line y = x + 2.

DRAW the circle and the line y = x + 2:


Only the blue portion of the circle lies above the line y = x+2.
Since the blue portion is equal to 1/4 of the circle, the probability that a point on the circle will satisfy the constraint that y>x+2 is 1/4.

Similar problems that are a bit more GMAT-friendly:
https://www.beatthegmat.com/probability-t90694.html
https://www.beatthegmat.com/coordinate-g ... 22193.html