Question

Line M is tangent to a circle, which is centered on point (3, 4). Does Line M run through point (6, 6)?
1. Line M runs through point (-8, 6)
2. Line M is tangent to the circle at point (3, 6)

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

Hint:

For geometry questions on the coordinate plane, always visualize or sketch the points and lines. The fact that the radius from (3,4) to (3,6) is vertical immediately tells you the tangent must be horizontal, which simplifies the problem immensely.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a data sufficiency question in coordinate geometry. We need to determine if a specific point, (6, 6), lies on a tangent line M. A line's equation must be uniquely determined to check if a point lies on it. The key property of a tangent line is that it is perpendicular to the radius at the point of tangency.

Step 2: Key Formula or Approach:
- The center of the circle is C = (3, 4).
- A line is uniquely defined by two distinct points or one point and a slope.
- The slope of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
- Two lines are perpendicular if the product of their slopes is -1 (i.e., \(m_1 \times m_2 = -1\)), or if one is horizontal (slope=0) and the other is vertical (slope is undefined).

Step 3: Detailed Explanation:
Analyzing Statement (1): Line M runs through point (-8, 6)
This statement gives us one point on line M. However, knowing just one point is not enough to define a line. There are infinitely many lines passing through the point (-8, 6). Some of these lines could be tangent to the given circle, while others are not. Even if we consider only the lines through (-8,6) that are tangent to the circle, there would be two such lines. We don't have enough information to find the unique equation of line M.
Therefore, Statement (1) ALONE is not sufficient.
Analyzing Statement (2): Line M is tangent to the circle at point (3, 6)
This statement gives us the point of tangency, T = (3, 6).
The center of the circle is C = (3, 4).
The radius CT connects the center to the point of tangency.
Let's find the slope of the radius CT:
\[ m_{radius} = \frac{6 - 4}{3 - 3} = \frac{2}{0} \] The slope is undefined, which means the radius CT is a vertical line.
The tangent line M is perpendicular to the radius at the point of tangency. A line perpendicular to a vertical line must be a horizontal line.
A horizontal line has the equation \(y = \text{constant}\).
Since line M passes through the point of tangency (3, 6), the y-coordinate for every point on the line must be 6.
So, the equation of line M is \(y = 6\).
The question is: Does Line M run through point (6, 6)?
We check if the point (6, 6) satisfies the equation \(y = 6\). Yes, its y-coordinate is 6.
This gives a definitive "Yes" answer to the question.
Therefore, Statement (2) ALONE is sufficient.

Step 4: Final Answer:
Statement (2) alone is sufficient to answer the question, but statement (1) alone is not. Therefore, the correct option is (D).