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

Question

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.

Bunuel · Accepted Answer

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:

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:

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.

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honchos · Answer

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.

Bunuel · Answer

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

honchos · Answer

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

Bunuel · Answer

We are NOT asked whether the slope is negative, we are asked whether the slope is less than 1.

honchos · Answer

Got it my bad, misinterpreted the question.

pradeepram · Answer

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 ?

Bunuel · Answer

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    positive    coordinates".

pradeepram · Answer

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 ?

abhinav11 · Answer

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

amitbharadwaj7 · Answer

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.

bsahil · Answer

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

anu1706 · Answer

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...

anu1706 · Answer

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….

AIMGMAT770 · Answer

Is there any other way of doing this question.When I'm considering both together.I am not able to confidently visualize it.

Pranjal3107 · Answer

I also have the same doubts. Can someone please explain ?

Sneha2021 · Answer

I will try to explain why (1)+(2) is sufficient 
Let take a,b coordinate of the center of the circle
If the line has slope = 1, this means a=b (slope = b/a). The only case when a=b when the circle touches both x and y axis. 
To satisfy both statements, the circle must have two x-intercepts, and must either be tangent to the y-axis or not intersect the y-axis at all. 
1st Case - In which the circle is tangent to the y-axis—the value of a is equal to the circle’s radius, but the value of b is less than the radius. That is, a > b so the line’s slope b/a must be less than 1.
2nd Case - a is greater than the circle’s radius, while b is still less than that radius. Therefore, the slope b/a is also less than 1 in this case.
The slope has to be less than 1 in all cases satisfying both statements.

frankgraves · Answer

I just got it, "slope less than 1" means the same with "gradient of the line is less than that of y=x".
Thanks  Bunuel  !

ChiaLee · Answer

Hi Bunuel!

Can you please explain what these highlighted parts mean? How do you come to these conclusions? I am unable to solve this question.

Thanks.

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:

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

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.

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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615 to 715 in 15 Days (Ria's Strategies)

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