GMAT Questions | Coordinate Geometry Q4

Distance between center of circle and a line | Data Sufficiency

The question given below is a GMAT data sufficiency question in coordinate geometry. Concept: Finding the distance between a point (center of the circle) and a straight line. A GMAT hard math question - GMAT 700 to 750 level data sufficiency question in co geo.

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This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  3. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  4. EACH statement ALONE is sufficient to answer the question asked.
  5. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Numbers

All numbers used are real numbers.

Figures

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Note

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 4: Does the line x + y = 6 intersect or touch the circle C with radius 5 units?

Statement 1: Center of the circle lies in the third quadrant.
Statement 2: Point (-4, -4) does not lie inside the circle.


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Explanatory Answer | GMAT Coordinate Geometry DS

Step 1: Decode the Question Stem

Q1. What kind of an answer will the question fetch?
The question is a "DOES" question. For "Be verb" questions, the answer is "YES" or "NO".

Q2. When will the line intersect or touch the circle?
The line will either intersect or touch the circle if the distance between the center of the circle and any point on the line is less than or equal to the radius of the circle.

Q3. What information do we have about the line from the question stem?
The line is a negative sloping line passing through the II, IV, and I quadrant.

Approach: Counter Example
Let us a pick a point on the line, say A(3, 3) and use the information in the statement(s) to find one scenario where the line cuts or touches the circle and in another where the line lies outside the circle.


Step 2: Evaluate Statement 1 ALONE

Statement 1: The center of the circle lies in the third quadrant.

Let us evaluate two scenarios. The center could be at O1(-0.5, -0.5) or could be at O2(-10, -10).

Case 1: If the center is at O1(-0.5, -0.5), the distance between O1A is less than 5 units, the radius of the circle. So, the line will intersect with the circle.

Case 2: On the other hand if the center is at O2(-10, -10), then the distance between O2A will be greater than 5 units, the radius of the circle. So, the line will neither touch nor intersect with the circle.

We are NOT able to find a conclusive answer to the question using Statement 1.
Hence, statement 1 alone is NOT sufficient.
Eliminate answer options A and D.
Choices narrow down to B, C, or E.


Step 3: Evaluate Statement 2 ALONE

Statement 2: Point (-4, -4) does not lie inside the circle.

What does statement 2 mean?
The distance between the center of the circle and (-4, -4) is more than 5 units. Let us look at two scenarios.

Case 1: The center of the circle could be at (0, 0) and point (-4, -4) will lie outside the circle.
The distance between the center of the circle and point A(3, 3) is less than 5 units. Hence, the line will intersect with the circle.

Case 2: Conversely, the center of the circle could be at (-10, -10). Point (-4, -4) will still lie outside the circle and the distance between point A(3, 3) and the center (-10, -10) will be more than 5 units.
Hence, the line will neither touch nor intersect with the circle.

We are unable to answer the question with a DEFNITE YES/NO using Statement 2.
Hence, statement 2 alone is not sufficient.
Eliminate answer option B.


Step 4: Evaluate Statements TOGETHER

Statement 1: Center of the circle lies in the third quadrant.
Statement 2: Point (-4, -4) does not lie inside the circle.

Center of the circle lies in the third quadrant and Point (-4, -4) does not lie inside the circle.

If we could prove that we can find two instances - one in which all the conditions stated in the two statements are satisfied and the line either touches or intersects with the circle and in another one in which all the conditions are satisfied and the line does not either touch or intersect with the circle, then we can say that the data is insufficient. If we are not able to find such instances, then the data will be sufficient.

Consider these two options for the center of the circle.
The center could be at O1(-10, -10) or could be at O2(-0.1, -0.1). Both O1 and O2 are III quadrant points.

Case 1: The distance between O1(-10, -10) and (-4, -4) is more than 5 units and that between O1(-10, -10) and (3, 3) is also more than 5 units. Hence, all conditions stated in the two statements are satisfied; the line does not intersect or touch the circle.

Case 2: As the result from case 1 turns out that the line does not touch or intersect the circle, our intent is to find a data point where the line will either touch or intersect with the circle.

We need to find a coordinate for the center of the circle such that its distance from (-4, -4) is greater than the radius 5 and its distance from a point on the line (3, 3) is less than 5 units and it lies in the 3rd quadrant.

Careful consideration will make it evident that the point should be a 3rd quadrant point that is really close to the origin. (-0.1, -0.1) clearly fits the bill.

The distance between O2(-0.1, -0.1) and (-4, -4) is more than 5 units and that between O2(-0.1, -0.1) and (3, 3) is less than 5 units. Hence, all conditions stated in the two statements are satisfied and the line intersects with the circle.

We have managed to find a counter example even after combining the two statements.

Hence, statements together are not sufficient.
Eliminate answer option C.

Choice E is the correct answer.



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