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

Circle C and line K lie in the XY plane. If circle C is centered at the origin and has a radius 1, does line K intersect circle C?

The best way to solve this question would be to visualize/draw it.

No matter what the slope is, it’s possible for line not to cross the circle as the x intercept can be + infinite.

(1) The X-Intercept of line k is greater than 1 --> Just says that X-intercept is to the right of the circle. Not sufficient
(2) The slope of line k is -1/10 --> Just says that slope is negative -1/10 --> line is just going down. Not sufficient.

(1)+(2) As we don't know exact intercept of line and X-axis we can not determine whether line intersects the circle or not. Not sufficient.

To elaborate more: we can draw infinitely many parallel lines with X-intercept more than 1 and slope -1/10, some will intersect the circle (for example line with X-intercept 1.1) and some not (for example line with X-intercept 1,000,000). Check the image below for two possible scenarios: blue line (with the slope of -1/10 and the x-Intercept greater than 1) intersects the circle while the red line (also with the slope of -1/10 and the x-Intercept greater than 1) does not.



Answer: E.

For more on this issue check Coordinate Geometry Chapter of Math Book: https://gmatclub.com/forum/math-coordina ... 87652.html

Hope it helps.


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Look at the diagrams.

Statement 1: If x intercept > 1, the line can be any of the following (and can be drawn in many more ways)


Statement 2: If slope = -1/10, the line can be drawn in any of the following ways. (and many more)


Using both together: Look at the diagrams above. Both have x intercepts greater than 1 and slope = -1/10. In one case, it will intersect the circle, in the other case, it will not. SO both statements together are not sufficient. Answer (E).
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Thank you for the explanation....And the Math book is really helpfull.

Using statement 1: If the X-intercept of line K is greater than 1, it may or not intersect circle C. For example, if the line is x=3 then it does not intersect the circle C, but if the line is y = -x/4 + 1/2 then it has x-intercept = 2 (>1) and does intersect the circle C. Therefore statement (1) is insufficient.

Using statement 2: If the slope of the line is -1/10, then it may or may not intersect the circle C. For example, y= -x/10 + 2 does not intersect the circle, but y = -x/10 + 0.5 does intersect the circle, though both have a slope of -1/10. Therefore statement 2 is insufficient to answer the question.

Combining statements 1 and 2, the statements together are still insufficient to answer the question. For example, y = -x/10 + 0.5 has an x-intercept greater than 1 and a slope of -1/10, but it does intersect the circle C. However, y = -x/10 + 2 also has an x-intercept greater than 1 and a slope of -1/10, but it does not intersect the circle C.

The answer is (E).

Draw some diagrams to figure it out. Even using both statements, you get two cases - one in which K intersects C and another in which it doesn't.


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a. y=mx+c so for x= -c/m where m can be 1 or -1. c can be 0.1 or 100.

hence a is POE thus D is POE too.

similarly for b too 'c' can have any such values as a.

thus POE.

a+b

y= -0.1x +c meaning x= c/0.1 > 1 thus c > 0.1
hence y intercept can be 0.1 with a slope of -0.1
or 100 with the same slope. not sufficient.

E it is.

E it is. The diagramming method as shown by Karishma worked best for me.

Statement 1 is insufficient as it gives only x intercept. A line with an x-intercept > 1 may or may not pass through the circle given. INSUFFICIENT.

Statement 2 gives a negative slope. The line is decreasing as it goes from left to right but it does not give an indication whether the line intersects the circle or not. INSUFFICIENT.

Even taken together, two lines can be drawn both with x-intercept more than one and with given negative slope. One of the lines may intersect the circle and the other may not. INSUFFICIENT.

Thus the answer is E.

We can see that the circle is centered at origin with radius 1 and has the equation x^2 + y^2 = 1

Using statement (1): If the x-intercept of line k is greater than 1, it may or may not intersect the circle. The line can have any slope and any x-intercept and not all of such lines will intersect with the circle. Insufficient.

Using statement (2): Just knowing the slope without any point that the line passes through is insufficient as with a particular slope infinite lines can be drawn that do and do not intercept the circle. Insufficient.

Combining statement (1) and (2), we get a definite equation of one unique line as we know the x-intercept and the slope. Therefore we can conclude that this line will either intersect the circle or will not. Sufficient.

(C) is the answer.

Responding to a pm:



How do you figure that slope is negative from statement 1?

Slope = - y intercept/x intercept
So you get
m = -c/x (correct)
x = -c/m > 1

Here is my problem: why do you say that m must be negative? c could be negative instead.

Though using both statements, we know that x intercept is greater than 1 and the slope is negative.
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(1) The x-intercept of line K is greater than 1 implies that the x-intercept of line K is towards the right of circle C but this does not imply if line K intersects circle C; NOT sufficient.

(2) The slope of line K is -1/10 implies line K goes down left to right, but again this does not imply if line K intersects circle C; NOT sufficient.

Combining (1) and (2), we do not know the exact intercept of line K, and we cannot find if line K intersects circle C or not; NOT sufficient.

The correct answer is E.
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we do not to draw many things.

if k is perpendicular to x axis , k must be paralel to y axis. k have negative slope this mean k can not parallel to y, so if x intercept is greater than 1, k can cut the circle.

if we move k to the right, k can not cut circle

Great explanation Bunuel and VeritasKarishma. But how can i visualize a slope of -1/10 on a line. Like how steep that line would be? is there any way?

Check this post first: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2010/1 ... he-graphs/

x axis has slope 0. y axis has infinite slope.
See the line which has slope 1 (green in the diagram in post). Between x axis and green line, slope goes from 0 to 1. Between green line and y axis, slope goes from 1 to infinite.
See the line which slope -1 (purple line in the diagram in post). Between x axis and purple line, slope goes from 0 to -1 (so -0.1 lies here). Between purple line and y axis, slope goes from -1 to -infinity.
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Thanks so much VeritasKarishma. That was very helpful

I had a sneaking suspicion that it was E for this very reason. But, I am still having trouble understanding (despite visual) why the larger the x-intercept is, the less likely the line will intersect the circle.

Or I guess maybe I will try to explain it myself.

Only in the case of a parallel line will you be sure that the line will intersect if the y-intercept is less than or equal to 1.
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