M20-12

Official Solution:

Set \(T\) consists of all points \((x, y)\) such that \(x^2 + y^2 = 1\). If point \((a, b)\) is randomly selected from set \(T\), what is the probability that \(b > a + 1\)?

A. \(\frac{1}{4}\)
B. \(\frac{1}{3}\)
C. \(\frac{1}{2}\)
D. \(\frac{3}{5}\)
E. \(\frac{2}{3}\)


Consider the diagram below:



The circle represented by the equation \(x^2+y^2 = 1\) is centered at the origin and has a radius of \(r=\sqrt{1}=1\).

So, set \(T\) is the circle itself (red curve).

The question is: if point \((a,b)\) is selected from set \(T\) at random, what is the probability that \(b > a + 1\)? All points \((a,b)\) that satisfy this condition (belong to \(T\) and have a y-coordinate greater than x-coordinate + 1) lie above the line \(y = x + 1\) (blue line). You can see that the portion of the circle which is above the line is \(\frac{1}{4}\) of the whole circumference, hence the probability \(P = \frac{1}{4}\).

Note that while Geometry is not tested on GMAT Focus, Coordinate Geometry is tested under the Functions and Graphing sections found in the Official Guide for GMAT Focus Edition.


Answer: A