Problem Solving

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

Note that, x² + y² = 1 represents a circle centered at (0, 0) with radius 1 on the coordinate plane.
And if (a, b) such a point such that b > (a + 1), (a, b) must lie above the the line y = x + 1 on the coordinate plane.

Refer to the diagram below,

T contains all the points that lies on the perimeter of the red circle. And the blue line represents the line y = x + 1.

We can see that for only 1/4 of the perimeter (the top left part in the second quadrant) the points are above the line y = x + 1.

So, required probability = 1/4

The correct answer is A.

chaithu_bunny wrote:

Anju@Gurome wrote:And if (a, b) such a point such that b > (a + 1), (a, b) must lie above the the line y = x + 1 on the coordinate plane.

Hi Anju. Can you explain the quoted line more clearly?

For the point (a, b), x-coordinate is a and y-coordinate is b.
So, if b > (a + 1), the y-coordinate of the point is greater than (x-coordinate of the point + 1)

Now, in general we can write this as y > (x + 1)
If you are not convinced, think like this : all the points for which y-coordinate of the point is equal to (x-coordinate of the point + 1), will lie on the straight line y = x + 1
So, all the points with y-coordinate greater than (x-coordinate of the point + 1) will be represented by the region y > (x + 1).
To put it simply, on the coordinate plane every linear equation is represented by a straight line whereas every linear inequality is represented by a region.

Now, the straight line y = (x + 1) will divide the coordinate plane in two regions.
One side of y = (x + 1) will represent the region y < (x + 1) while the other side will represent y > (x + 1).

To identify which side represent which region, check with the point (0, 0)
In this case, 0 < (0 + 1)
So, (0, 0) must be lying in the region y < (x + 1)
So, the region below the line y = (x + 1) represents y < (x + 1)
So, the region above the line y = (x + 1) represents y > (x + 1)

Hope that helps.

Note : For any problem, if (0, 0) lies on the deciding line, then pick some other simple point lies (1, 1) or (0, 1) or (1, 0) to determine which side represents which inequality.