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A circle is drawn on a coordinate plane. If a line is drawn

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A circle is drawn on a coordinate plane. If a line is drawn

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A circle is drawn on a coordinate plane. If a line is drawn through the origin and the center of that circle, is the line’s slope less than 1?

(1) No point on the circle has a negative x-coordinate.

(2) The circle intersects the x-axis at two different positive coordinates

Please answer in detail.

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A circle is drawn on a coordinate plane. If a line is drawn through the origin and the center of that circle, is the line’s slope less than 1?

(1) No point on the circle has a negative x-coordinate.
(2) The circle intersects the x-axis at two different positive coordinates
This is how I approached the problem...

Q: We need to find if the slope of the line from the point (0, 0) to the center of a circle is LESS than 1 (let this be out target line)

St1:

No point on the circle has a negative x-coordinate.
This essentially means that the circle is on the right hand side of the Y Axis.
But we can have many Target lines satisfying the above but having slopes Greater than 1 (Green line), Equal to 1 (y = x) or Less than 1 (Blue line)



More than one possibility, thus INSUFFICIENT

St2:

Again we can have more than one possibility, thus INSUFFICIENT

St1+St2:
Now we are limited to the right hand side of the y axis. As its mentioned that "No point has a negative x coordinate" the least value can be x = 0. Only one point can have this value and the y axis would act as a tangent:


Now, a circle on the line y = x which satisfies the limitation "x should be non negative" has two axes as the two tangents:


BUT we need the circle to intersect two points on the x axis, Thus we would have to move it DOWN:


Thus both are sufficient together, Answer C

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Hi coolanujdel,

mevicks has offered a great, thorough explanation of how to answer this question. I'll add that on most graphing questions (as well as most geometry questions in general), you'll find it helpful to physically draw whatever you're working on (and if it's a DS question, then you'll probably need to create several drawings). This is a great way to be sure that your work is correct.

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coolanujdel wrote:
A circle is drawn on a coordinate plane. If a line is drawn through the origin and the center of that circle, is the line’s slope less than 1?

(1) No point on the circle has a negative x-coordinate.

(2) The circle intersects the x-axis at two different positive coordinates

Please answer in detail.
mevicks work is great and scrupulous.

To answer this, we need to know the nature of the coordinates of the center of this circle.

(1) This means that no part of the circle lies in the second or third quadrant. In other words, the center is one of these forms: (+, +), (+, -), or (+, 0). Let’s plug in for the center. If the center is (3, 2), slope of the mentioned line is 2/3 or less than 1, but, if the center is (2, 3), then slope of the mentioned line is 3/2 or more than 1. Hence insufficient. A and D out, B, C, E to focus on.

(2) This doesn't fix the nature of the coordinates of the center of this circle. The center could be below, above or on the line y = x to get different conclusions about the slope of the mentioned line. Choice B out, our answer is among C and E only.

When the two statements are taken together, it limits us to the right side of the y-axis only. The slope of the mentioned line is 1 if the circle is in first quadrant with the axes tangent to it. But, since the circle has to intersect the x-axis at two different positive coordinates, we need to take it little or more down the x-axis thus making sure that the slope of the mentioned line is less than 1. That’s sufficient, hence C is the correct answer!

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Sanjeev K Saxena
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sanju09 wrote:
coolanujdel wrote:
A circle is drawn on a coordinate plane. If a line is drawn through the origin and the center of that circle, is the line’s slope less than 1?

(1) No point on the circle has a negative x-coordinate.

(2) The circle intersects the x-axis at two different positive coordinates

Please answer in detail.
mevicks work is great and scrupulous.

To answer this, we need to know the nature of the coordinates of the center of this circle.

(1) This means that no part of the circle lies in the second or third quadrant. In other words, the center is one of these forms: (+, +), (+, -), or (+, 0). Let’s plug in for the center. If the center is (3, 2), slope of the mentioned line is 2/3 or less than 1, but, if the center is (2, 3), then slope of the mentioned line is 3/2 or more than 1. Hence insufficient. A and D out, B, C, E to focus on.

(2) This doesn't fix the nature of the coordinates of the center of this circle. The center could be below, above or on the line y = x to get different conclusions about the slope of the mentioned line. Choice B out, our answer is among C and E only.

When the two statements are taken together, it limits us to the right side of the y-axis only. The slope of the mentioned line is 1 if the circle is in first quadrant with the axes tangent to it. But, since the circle has to intersect the x-axis at two different positive coordinates, we need to take it little or more down the x-axis thus making sure that the slope of the mentioned line is less than 1. That’s sufficient, hence C is the correct answer!
Can you please tell how to approach these problems when the concept is not in any book.I am really struggling for questions level 600 and above.

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coolanujdel wrote:
Can you please tell how to approach these problems when the concept is not in any book.I am really struggling for questions level 600 and above.
Certain basic concepts are must to know, and certain are not so must, especially whenever plug in works, we might not need to remember a missing concept and can still do it right, but sometimes it becomes must to know.

Good idea is to take time and go through the books like The Princeton Review's Quantitative Review for the GMAT,Version 9.0, where you'll learn most of the necessary basic concepts that are essential to know for a test taker.

Remember, no concept can be learnt without thorough practice over it,

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The mind is everything. What you think you become. –Lord Buddha



Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

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