Coordinate Geometry DS - Lines and Circles

A data sufficiency question in coordinate geometry

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

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

Correct Answer is Choice E. The data is INSUFFICIENT.

Explanatory Answer

The line will 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.

Let us a pick a point on the line - say A(3, 3).

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

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), then 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.

Hence, from statement 1 we cannot answer the question. Data INSUFFICIENT.

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

The distance between the center of the circle and (-4, -4) is more than 5 units.

Case 1: The center of the circle could be at (0, 0) and point (-4, -4) will lie outside the circle. However, the distance between the center 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.

Hence, from statement 2 we cannot answer the question. Data INSUFFICIENT.

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

Consider the two following options for the center of the circle.

The center could be at O1(-0.1, -0.1) or could be at O2(-10, -10).
Case 1: The distance between  O1(-0.1, -0.1) and (-4, -4) is more than 5 units and that between O1(-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.
Case 2: The distance between  O2(-10, -10) and (-4, -4) is more than 5 units and that between O2(-10, -10) and (3, 3) is also more than 5 units. Hence, all conditions stated in the two statements are satisfied - but the line does not intersect or touch the circle.

Therefore, using the statements independently or together we will not be able to answer the question.

Choice E is the answer.




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