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

Question

Does the line x + y = 6 intersect or touch the 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.

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yashikaaggarwal · Answer

(1) Center of the circle lies in the third quadrant.
There are infinite number of circles possible with the center lying in 3rd quadrant.
Which may or may not touch or intersect line x+y=6
(Insufficient)

(2) Point (-4, -4) does not lie inside the circle.
The distance between point (-4,-4) and origin is > 5
So the center of the circle could be in any quadrant. Considering that, the circle could be tangent to line x+y=6, Bisected by line or could be far away.
(Insufficient).

Combining both,
Center is in 3rd quadrant.
But the distance between point (-4,-4) and origin is > 5
Plus the center could be inclined from any point in 3rd quadrant.
No 1 possible answer.
(Insufficient)

Answer is E

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solikon · Answer

From the statement prompt, we know that a slope of a line is -1 (Given the equation (x + y = 6)).

Statement A  tells us nothing particularly and the answer could be 'YES' and 'NO'. Options A and D are out

Statement B  Also answer to this prompt could be YES' and 'NO' too. Options B and C are out, Option  E  left.

In case that the circle center coordinates would be (-1, -0.1) answer would be 'YES' as the circle would be crossed by the line with a slope -1 passing through the point with coordinates (-4,-4).
In case that the circle center coordinates would be (-1, -15) answer would be 'NO' as the circle wouldn't be crossed by the line with a slope -1 passing through the point with coordinates (-4,-4).

Hence, the correct answer is  E .

P.S.: For a better understanding please see a picture.

GCMEMBER · Answer

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

The line x + y = 6 passes through (6,0) and (0,6)

The mid-point of these vertices is (3,3), which lies at the minimum distance to this line from the origin
Minimum Distance from origin to line= √18

(1) Center of the circle lies in the third quadrant.
Since minimum distance from origin to the line is √18 and maximum distance from the centre of the circle to the line can be √25, a circle may or may not touch the line.

Not sufficient

(2) Point (-4, -4) does not lie inside the circle.
Draw a circle at origin with radius 5. It will intersect the line. 
Draw a circle at (-10,-10) with radius 5. It will not intersect the line. 
Not sufficient

(1) + (2)
The distance between (3,3) and (-4,-4) is √98
While the diameter of the circle is √100
So any circle drawn between these points will either include (-4,4) inside it or intersect the line or both.
To have a circle with radius 5, with centre in the third quadrant and which excludes (-4,4), it must intersect the line l.

Sufficient

Option C

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AndrewN · Answer

I have drafted a graph to help others visualize why the answer should be (E). The red line shows, well, the graph of the line given in the setup. The blue circle shows what could happen if the center of the circle were close to the origin, while the green circle indicates what could happen if the center of the circle were further away from the origin. You will note that each circle conforms to the information from the statements:

1) The center of the circle does, in fact, lie in quadrant III.
2) The point (-4, -4) does NOT lie within the circle.

Attachment:

Screen Shot 2021-02-10 at 14.44.34.png [ 118.93 KiB | Viewed 5211 times ]

I hope this helps. Good luck with your studies, everyone.

- Andrew

CEdward · Answer

I hope I am not being too microscopic here, but for the purposes of the GMAT...if we get something like statement 1, then would circles that fall in quadrant 3 and perhaps others (e.g. blue circle) count? Or does it have to be FULLY in quadrant 3?

Edit: I read your response again. So what matters is the statements are satisfied.

Edit: I read your response again. So what matters is the statements are satisfied.

AndrewN · Answer

Yes,  CEdward , the point is that you  should  think microscopically. In this case, one of the extremes—I teach my students all the time to consider extremes within the given information—reveals a flaw in the reasoning that the  circle , NOT the  center of the circle  only, must lie within the third quadrant. You can work within the information given in the problem, as well as the information given in each statement, but anything else, apart from pertinent outside mathematical knowledge, is speculation or assumption on your part, and that can get you into trouble really fast on DS questions in particular.

Yes, consider whether each statement allows you to answer the question being asked, nothing more and nothing less. Good luck with your studies. (I see you have become quite active in the forum. I wonder if your test is coming up soon.)

- Andrew

Anchal270789 · Answer

Your graph is quite self explanatory. However when we encounter such qs in gmat exam, we can cross out choices A, B and D..But in order to check option C, Im unable to make sure whether I can draw a circle with centre in quad III, having (-4,-4) outside the circle and the circle touching the Line. The only way which I could think of was finding distance between 0,0 and -4,-4 and making sure it is more than radius 5

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AndrewN · Answer

Hello,  Anchal270789 . There is nothing wrong with checking such a distance, but I would recommend an easy approach even so. Since (-4, -4) presents x- and y-coordinates that are equidistant from the origin, you can quickly picture a 45-45-90 right triangle in which sides will be x-x-x√2, respectively. In this case,  x  is 4, so the distance is 4√2. The square root of 2 is about 1.4—a figure that can be useful to memorize for the GMAT™—and 1.4 times 4 is 5.6. That 0.6 certainly gives you some wiggle room for where the center of the circle may be. On the positive side, in quadrant I, the line passes through the point (3, 3), which would be a distance of only 3√2 units from the origin. 3 times 1.4 is 4.2. It is thus quite conceivable that a circle with its center in quadrant III  could  touch the line and extend beyond it. (On open-ended questions such as this one, you can effectively treat the origin as the center of the circle since the exact center could be something like (-0.00000000000000000001, -0.00000000000000000001).)

I hope that helps. Good luck with your studies.

- Andrew

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

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