Question

If X and Y are points in a plane and X lies inside the circle C with center O and radius 2, does Y lie inside circle C?
(1) The length of line segment XY is 3.
(2) The length of line segment OY is 1.5.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

For geometry-based Data Sufficiency questions, especially those involving inequalities, always try to construct counterexamples. If you can find one case where the answer is "Yes" and another where it's "No", the statement is not sufficient.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We are given a circle C with center O and radius 2. 
A point P is inside the circle if its distance from the center, OP, is less than the radius. 
We are given that point X is inside the circle, which means the distance OX \(< 2\). 
The question asks: Is point Y inside the circle? This is equivalent to asking: Is the distance OY \(< 2\)? 
Step 2: Detailed Explanation: 
Analyze Statement (1): The length of line segment XY is 3. 
This means the distance between X and Y is 3. We can use the triangle inequality for points O, X, and Y: OY \(\le\) OX + XY. Since we know OX \(< 2\) and XY = 3, we have: \[ \text{OY}<2 + 3 = 5 \] This tells us that OY is less than 5, but it doesn't tell us if OY is less than 2. Let's test two scenarios: 

Scenario A (Y is outside): Let O be at the origin (0,0). Let X be at (1,0). OX = 1, which is \(< 2\). If Y is at (4,0), then the distance XY = \(|4-1| = 3\). The distance OY = 4, which is not \(< 2\). In this case, Y is outside the circle. The answer to the question is ""No"". 
Scenario B (Y is inside): Let O be at (0,0). Let X be at (1.5, 0). OX = 1.5, which is \(< 2\). If Y is at (-1.5, 0), then the distance XY = \(|1.5 - (-1.5)| = 3\). The distance OY = 1.5, which is \(< 2\). In this case, Y is inside the circle. The answer to the question is ""Yes"". 
Since we can get both ""Yes"" and ""No"" answers, statement (1) is not sufficient. 
Analyze Statement (2): The length of line segment OY is 1.5. 
This statement directly gives us the distance of point Y from the center O. We need to determine if Y is inside the circle, which means we need to know if OY \(< 2\). 
The statement says OY = 1.5. Is \(1.5<2\)? Yes. This gives us a definitive ""Yes"" answer to the question. Therefore, statement (2) is sufficient. 
Step 3: Final Answer: 
Statement (2) alone is sufficient, but statement (1) alone is not.