Question

In the xy-coordinate system, the point (x, y) lies on the circle with equation \(x^2 + y^2 = 1\).
Column A: \(x + y\)
Column B: 1.01

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

For problems involving points on a circle, testing key points like the intersections with the axes (\((1,0), (0,1), (-1,0), (0,-1)\)) and the points on the lines \(y=x\) and \(y=-x\) can quickly reveal the range of possible values for expressions like \(x+y\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are given that a point \((x,y)\) is on the unit circle (a circle centered at the origin with radius 1). We need to compare the sum of its coordinates, \(x+y\), with the value 1.01. This requires understanding the range of possible values for \(x+y\).
Step 2: Detailed Explanation:
The value of \(x+y\) changes depending on where the point \((x,y)\) is on the circle. Let's test some points.
Case 1: Consider the point where the circle intersects the positive x-axis. Here, the coordinates are \((1, 0)\).
For this point, \(x+y = 1+0 = 1\).
In this case, Column A (1) is less than Column B (1.01).
Case 2: Consider the point in the first quadrant where \(x=y\).
Substitute \(y=x\) into the circle equation: \(x^2 + x^2 = 1 \implies 2x^2 = 1 \implies x^2 = 1/2 \implies x = 1/\sqrt{2}\).
So the point is \((1/\sqrt{2}, 1/\sqrt{2})\).
For this point, \(x+y = 1/\sqrt{2} + 1/\sqrt{2} = 2/\sqrt{2} = \sqrt{2}\).
The value of \(\sqrt{2}\) is approximately 1.414.
In this case, Column A (\(\approx 1.414\)) is greater than Column B (1.01).
Since we have found one case where Column A is less than Column B, and another case where Column A is greater than Column B, the relationship is not fixed.
Step 3: Final Answer:
The relationship cannot be determined from the information given.