Question

The points that satisfy the system $x+y=1,$ $x^2+y^2<25,$ where the symbol “ $<$ ” means “less than,” constitute the following set:

only two points

an arc of a circle

a straight line segment not including the end-points

a straight line segment including the end-points

a single point

Answer: C

Concepts:coordinate geometry circle inequality

Difficulty rating: 1300

Small Hint:

Interpret $x^2+y^2<25$ as the interior of a circle

Big Hint:

Intersect that open disk with the line $x+y=1$

Solution:

The inequality describes the open disk inside the circle of radius $5$ centered at the origin. The line $x+y=1$ crosses the circle in two points. Its portion inside the disk is the segment between those points, but the strict inequality excludes the endpoints.

Thus, the correct answer is C.

1957 AMC 12 Problem 34

Answer: C

Difficulty rating: 1300

Small Hint:

Big Hint:

Solution:

Problem 34 in Other Years