Circle C and line K lie in the XY plane. If circle C is centered at the origin and has...

Understanding the Question

Circle C is centered at the origin \((0,0)\) with radius 1. We need to determine: Does line K intersect circle C?

This is a yes/no question. To be sufficient, we need to definitively answer either "yes, line K always intersects" or "no, line K never intersects" based on the given information.

Key geometric insight: A line intersects a circle when the perpendicular distance from the circle's center to the line is less than or equal to the circle's radius. Since our circle has radius 1 and is centered at the origin, line K intersects the circle if and only if the distance from \((0,0)\) to line K is \(\leq 1\).

Analyzing Statement 1

Statement 1 tells us: The x-intercept of line K is greater than 1.

This means line K crosses the x-axis at some point to the right of \(\mathrm{x} = 1\). However, this tells us nothing about the line's slope (direction).

Let's visualize different possibilities:

• Imagine a line crossing at \(\mathrm{x} = 2\) with a horizontal slope (slope = 0). This line would be \(\mathrm{y} = 0\), which passes right through the origin and definitely intersects the circle.
• Now imagine the same line crossing at \(\mathrm{x} = 2\) but with a very steep negative slope (like -10). This line would stay far from the origin and might miss the circle entirely.

Since we can construct scenarios where the line intersects the circle AND scenarios where it doesn't, Statement 1 is NOT sufficient.

[STOP - Statement 1 is NOT Sufficient!]

This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: The slope of line K is -1/10.

This gives us the line's direction but not its position. All lines with slope -1/10 are parallel to each other, but they can be at different distances from the origin.

Let's think about different parallel lines:

• A line with slope -1/10 passing through the origin would definitely intersect the circle (distance = 0).
• A line with slope -1/10 passing through the point \((0, 100)\) would be far above the circle and definitely miss it.
• Somewhere in between, there are lines that just graze the circle's edge.

Since different positions of the line (all with the same slope) lead to different answers about intersection, Statement 2 is NOT sufficient.

[STOP - Statement 2 is NOT Sufficient!]

This eliminates choice B.

Combining Statements

From both statements together, we know:

• The x-intercept is greater than 1
• The slope is -1/10

With both pieces of information, we can write any line as: \(\mathrm{y} = -1/10(\mathrm{x} - \mathrm{a})\) where \(\mathrm{a} > 1\) is the x-intercept.

Even with both constraints, the line's exact position still varies depending on the value of 'a'.

Let's test extreme cases:

• If the x-intercept is \(\mathrm{a} = 1.1\) (just slightly more than 1), the line is very close to the circle and likely intersects it.
• If the x-intercept is \(\mathrm{a} = 100\) (far to the right), the line stays far from the origin and likely misses the circle.

To visualize this better: when the x-intercept increases, the line moves further right while maintaining its -1/10 slope, creating a "sliding effect" that takes the line farther from the origin.

Since we can construct scenarios that satisfy both constraints but give different answers to whether the line intersects the circle, the statements together are NOT sufficient.

[STOP - Combined Statements are NOT Sufficient!]

The Answer: E

The statements together are not sufficient because even knowing both the x-intercept constraint and the slope, we can position the line to either intersect or miss the circle depending on the specific x-intercept value chosen.

Answer Choice E: "The statements together are not sufficient."