Problem Solving

Hello,

Can you please assist with this:

In a coordinate system, how many points (x,y) simultaneously satisfy the conditions
|x| + |y| ≤ 1 and x^2 + y^2 = 1?

(A) exactly one
(B) exactly two
(C) exactly three
(D) exactly four
(E) infinite points

OA: D

Thanks,
Sri

gmattesttaker2 wrote:Hello,

Can you please assist with this:

In a coordinate system, how many points (x,y) simultaneously satisfy the conditions
|x| + |y| ≤ 1 and x^2 + y^2 = 1?

(A) exactly one
(B) exactly two
(C) exactly three
(D) exactly four
(E) infinite points

OA: D

Thanks,
Sri

Some points that satisfy |x| + |y| ≤ 1
(0,1), (0.25,0.75), (0.5,0.5), (0.75,0.25), (1,0), (0.5,-0.5), (0,-1), (-0.5,-0.5), (-1,0), (-0.5,0.5)

On plotting, we realize that this set of points forms a square with vertices (0,1), (1,0), (0,-1), (-1,0)

The equation x^2 + y^2 = 1 represents a circle with center at (0,0) and radius = 1.
The only points where the circle meets the above square are at the square's four vertices.

Choose D

Cheers

gmattesttaker2 wrote:Hello,

Can you please assist with this:

In a coordinate system, how many points (x,y) simultaneously satisfy the conditions
|x| + |y| ≤ 1 and x^2 + y^2 = 1?

(A) exactly one
(B) exactly two
(C) exactly three
(D) exactly four
(E) infinite points

OA: D

Thanks,
Sri

x² + y² = r² is the equation of circle centered at the origin.

Thus:
x² + y² = 1 is a circle centered at the origin with a radius of 1:

The 4 points shown -- (0,1), (1,0), (0,-1), and (-1,0) -- all satisfy the constraint that |x| + |y| ≤ 1.

If we choose ANY OTHER POINT on the circle, we get something like this:

Since the hypotenuse of the yielded triangle has a length of 1, the sum of the two legs -- |x| + |y| -- must be GREATER than 1.

Thus, only 4 points on circle x² + y² = 1 satisfy the constraint that |x| + |y| ≤ 1:
(0,1), (1,0), (0,-1), and (-1,0).

The correct answer is D.