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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  [#permalink]

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New post 15 Oct 2013, 22:17
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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.
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A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 16 Oct 2013, 02:02
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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. This implies that the center of the circle is either in I or IV quadrant AND the x coordinate of the center is greater than the radius of this circle. Still not sufficient. Consider the cases below:

Image

Not sufficient.

(2) The circle intersects the x-axis at two different positive coordinates. This implies that the distance from the center to the x-axis is less than the radius:

Image

Not sufficient.

(1)+(2) We know that the center is either in I or IV quadrant. We also know that the x coordinate of the center is greater than the radius of this circle and the distance from the center to the x-axis is less than the radius. This implies that the center is below line y=x, which means that the slope of a line passing origin and the center is less than 1. Sufficient.

Answer: C.

Attachment:
Untitled2.png
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Attachment:
Untitled.png
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post Updated on: 16 Oct 2013, 02:41
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I believe E should be the answer.

Y= mx + c C=0, since line will pass through 0,0

Y = MX

M depends upon the sign of Y, From (1) and (2), Y can be negative or positive, Not sure hence E.
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Originally posted by honchos on 16 Oct 2013, 02:35.
Last edited by honchos on 16 Oct 2013, 02:41, edited 1 time in total.
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 16 Oct 2013, 02:37
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post Updated on: 16 Oct 2013, 02:42
Bunuel wrote:
honchos wrote:
I believe E should be the answer.

Y= mx + c C=0, since line will pass through 0,0

Y = MX

M depends upon the sign of Y, From (1) and (2), Y can be negative or positive, Not sure hence E.


The question asks whether the slope is less than 1. The correct answer is C.


M = Y/X, X is always positive but Y may be Positive or negative, so slope may be +ve or -ve... Hence E

M: stands for line's slope
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Originally posted by honchos on 16 Oct 2013, 02:40.
Last edited by honchos on 16 Oct 2013, 02:42, edited 1 time in total.
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 16 Oct 2013, 02:42
honchos wrote:
Bunuel wrote:
honchos wrote:
I believe E should be the answer.

Y= mx + c C=0, since line will pass through 0,0

Y = MX

M depends upon the sign of Y, From (1) and (2), Y can be negative or positive, Not sure hence E.


The question asks whether the slope is less than 1. The correct answer is C.


M = Y/X, X is always positive but Y may be Positive or negative, so slope may be +ve or -ve... Hence E

[b]M[/b]: stands for line's slope


We are NOT asked whether the slope is negative, we are asked whether the slope is less than 1.
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 16 Oct 2013, 12:16
Got it my bad, misinterpreted the question.
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 25 Oct 2013, 04:00
Isn't it possible that the circle touches the origin and the radius and the center ( the center lies on the x axis and there is no y co-ordinate for it) of the circle is same ?
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 25 Oct 2013, 04:20
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pradeepram wrote:
Isn't it possible that the circle touches the origin and the radius and the center ( the center lies on the x axis and there is no y co-ordinate for it) of the circle is same ?


If center of the circle is on the x-axis the y-coordinate of the center is 0. In this case the slope of the line will be 0 (since the line would be horizontal), so still less than 1.

Also, the circle cannot pass through the origin, because we are told that "the circle intersects the x-axis at two different [u]positive[/u] coordinates".
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 25 Oct 2013, 04:36
Thanks Bunuel.

I considered this option for statement 1 alone. Since this is one of the possibilities, we could also take this into consideration for statement 1 right ?
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 26 Oct 2013, 04:46
Bunuel 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. This implies that the center of the circle is either in I or IV quadrant AND the x coordinate of the center is greater than the radius of this circle. Still not sufficient. Consider the cases below:
Attachment:
Untitled.png
Not sufficient.

(2) The circle intersects the x-axis at two different positive coordinates. This implies that the distance from the center to the x-axis is less than the radius:
Attachment:
Untitled2.png
Not sufficient.

(1)+(2) We know that the center is either in I or IV quadrant. We also know that the x coordinate of the center is greater than the radius of this circle and the distance from the center to the x-axis is less than the radius. This implies that the center is below line y=x, which means that the slope of a line passing origin and the center is less than 1. Sufficient.

Answer: C.


Hi Bunuel,

As always great explanation, Just one question you said as the center is below line y=x , means that slope of a line is less than 1 how do you come to that conclusion??

I am sorry I am weak in Co-ordinate geometry so pardon me if I am missing any basic Formulae for Slope ..

Regards,
Abhinav
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 27 Oct 2013, 05:56
abhinav11 wrote:
Bunuel 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. This implies that the center of the circle is either in I or IV quadrant AND the x coordinate of the center is greater than the radius of this circle. Still not sufficient. Consider the cases below:
Attachment:
Untitled.png
Not sufficient.

(2) The circle intersects the x-axis at two different positive coordinates. This implies that the distance from the center to the x-axis is less than the radius:
Attachment:
Untitled2.png
Not sufficient.

(1)+(2) We know that the center is either in I or IV quadrant. We also know that the x coordinate of the center is greater than the radius of this circle and the distance from the center to the x-axis is less than the radius. This implies that the center is below line y=x, which means that the slope of a line passing origin and the center is less than 1. Sufficient.

Answer: C.


Hi Bunuel,

As always great explanation, Just one question you said as the center is below line y=x , means that slope of a line is less than 1 how do you come to that conclusion??

I am sorry I am weak in Co-ordinate geometry so pardon me if I am missing any basic Formulae for Slope ..

Regards,
Abhinav


Dear Abhinav;

The standard equation of the line is y= mx + c, It is given that the line passes through origin, so coordinate will be (0, 0). To satisfy the condition given i.e. (0,0), you will find the modified equation as y= mx

now the question is asking whether m < 1.

if you assume m=1, you will find a straight line y = x, you will find this line bisects (45 deg) the first quadrant, try to solve the question by drawing, you will understand it much better. in equation y = x, each value of y is equal to x

Now coming to question, for m to be less than 1, if you shift the center of circle below the line y= x, then in new equation of line y=mx, for every value of y, you will have greater corresponding value of x, which indeed is the condition we are looking for y / x < 1.

Similarly, if you shift the center of circle above the line y=x, then in new equation of line y=mx, for every value of y, you will lower corresponding value of x.

If you did not get what I am trying to explain, try to draw this and you will understand.
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 27 Oct 2013, 06:47
Bunuel 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. This implies that the center of the circle is either in I or IV quadrant AND the x coordinate of the center is greater than the radius of this circle. Still not sufficient. Consider the cases below:
Attachment:
Untitled.png
Not sufficient.

(2) The circle intersects the x-axis at two different positive coordinates. This implies that the distance from the center to the x-axis is less than the radius:
Attachment:
Untitled2.png
Not sufficient.

(1)+(2) We know that the center is either in I or IV quadrant. We also know that the x coordinate of the center is greater than the radius of this circle and the distance from the center to the x-axis is less than the radius. This implies that the center is below line y=x, which means that the slope of a line passing origin and the center is less than 1. Sufficient.

Answer: C.



Hi Bunuel,

Please explain the point " the x coordinate of the center is greater than the radius of this circle", which you have derived from statement 1

Thanks,
Sahil
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 27 Oct 2013, 10:42
Bunuel 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. This implies that the center of the circle is either in I or IV quadrant AND the x coordinate of the center is greater than the radius of this circle. Still not sufficient. Consider the cases below:
Attachment:
Untitled.png
Not sufficient.

(2) The circle intersects the x-axis at two different positive coordinates. This implies that the distance from the center to the x-axis is less than the radius:
Attachment:
Untitled2.png
Not sufficient.

(1)+(2) We know that the center is either in I or IV quadrant. We also know that the x coordinate of the center is greater than the radius of this circle and the distance from the center to the x-axis is less than the radius. This implies that the center is below line y=x, which means that the slope of a line passing origin and the center is less than 1. Sufficient.

Answer: C.


Can this be proven algebracilly, Second I have not understood your statement "AND the x coordinate of the center is greater than the radius of this circle". and also this statement ". This implies that the distance from the center to the x-axis is less than the radius"
Please help and tell how to approach the problem as there will be no graphs available in the test...
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A circle is drawn on a coordinate plane. If a line is drawn.  [#permalink]

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New post 28 Oct 2013, 07:34
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.

Can this be proven algebraically, Second I have not understood your statement "AND the x coordinate of the center is greater than the radius of this circle". and also this statement ". This implies that the distance from the center to the x-axis is less than the radius"
Please help and tell how to approach the problem….
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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New post 09 Apr 2015, 00:21
Bunuel 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. This implies that the center of the circle is either in I or IV quadrant AND the x coordinate of the center is greater than the radius of this circle. Still not sufficient. Consider the cases below:
Attachment:
Untitled.png
Not sufficient.

(2) The circle intersects the x-axis at two different positive coordinates. This implies that the distance from the center to the x-axis is less than the radius:
Attachment:
Untitled2.png
Not sufficient.

(1)+(2) We know that the center is either in I or IV quadrant. We also know that the x coordinate of the center is greater than the radius of this circle and the distance from the center to the x-axis is less than the radius. This implies that the center is below line y=x, which means that the slope of a line passing origin and the center is less than 1. Sufficient.

Answer: C.


Is there any other way of doing this question.When I'm considering both together.I am not able to confidently visualize it.
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Re: A circle is drawn on a coordinate plane. If a line is drawn  [#permalink]

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Re: A circle is drawn on a coordinate plane. If a line is drawn   [#permalink] 08 Jul 2019, 03:58
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