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# A circle with center O and radius 5 is shown in the xy-plane

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Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

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20 Jul 2018, 09:29
GMATGuruNY wrote:
violetsplash wrote:
A circle with center O and radius 5 is shown in the xy-plane. Lines that intersect the circle in 2 points include which of the following ?

I. y = -x +1
II. y = 2x + 1
III. y = (1/2)x - 6

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II and III

Any line that passes through the interior of the circle will intersect the circle at two points.
The x-intercepts of the circle are -5 and 5.
The y-intercepts of the circle are -5 and 5.

I. y = -x + 1
II. y = 2x + 1
Each of these lines has a y-intercept between -5 and 5.
Thus, each line must pass through the interior of the circle.
Eliminate any answer choice that does not include both I and II (A, B and D).

III. y = (1/2)x - 6
Here, the y-intercept = -6.
To determine the x-intercept, substitute y=0 and solve for x:
0 = (1/2)x - 6
x=12.
Since the x-intercept = 12 and the y-intercept = -6, this line does not pass through the interior of the circle.
Eliminate any remaining answer choice that includes III (E).

Hi GMATGuruNY
In the two cases below, did you set Y = 0 to find Y-intercept and X = 0 to find X-intercept ? Do we need to test both Y and X intercept to detrmine wether question is correct or not ? thank you and have a nice weekend

I. y = -x + 1
II. y = 2x + 1
Each of these lines has a y-intercept between -5 and 5.
Thus, each line must pass through the interior of the circle.
Eliminate any answer choice that does not include both I and II (A, B and D).
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Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

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20 Jul 2018, 10:30
dave13 wrote:
Hi GMATGuruNY
In the two cases below, did you set Y = 0 to find Y-intercept and X = 0 to find X-intercept ? Do we need to test both Y and X intercept to detrmine wether question is correct or not ? thank you and have a nice weekend

I. y = -x + 1
II. y = 2x + 1
Each of these lines has a y-intercept between -5 and 5.
Thus, each line must pass through the interior of the circle.
Eliminate any answer choice that does not include both I and II (A, B and D).

It is not necessary to find the x-intercepts for these two lines.
Any line in the form y = mx + b has a y-intercept at (0, b).
Thus, each of the lines above has a y-intercept at (0, 1).
(0, 1) is within the interior of the circle.
Since each line has a y-intercept within the circle, each must pass through the interior of the circle and thus intersect the circle at two points.
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Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

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20 Jul 2018, 10:30
1
dave13 wrote:
Hi GMATGuruNY
In the two cases below, did you set Y = 0 to find Y-intercept and X = 0 to find X-intercept ? Do we need to test both Y and X intercept to detrmine wether question is correct or not ? thank you and have a nice weekend

I. y = -x + 1
II. y = 2x + 1
Each of these lines has a y-intercept between -5 and 5.
Thus, each line must pass through the interior of the circle.
Eliminate any answer choice that does not include both I and II (A, B and D).

It is not necessary to find the x-intercepts for these two lines.
Any line in the form y = mx + b has a y-intercept at (0, b).
Thus, each of the lines above has a y-intercept at (0, 1).
(0, 1) is within the interior of the circle.
Since each line has a y-intercept within the circle, each must pass through the interior of the circle and thus intersect the circle at two points.
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Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

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22 Jul 2018, 09:49
This question can be solved by finding the x and y intercepts for each of the three lines and comparing them to the x and y intercepts of the circle. Once could easily figure out whether the line is intersecting the circle or not. Pretty straightforward stuff!!
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Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

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03 Aug 2018, 03:17
KarishmaB wrote:
Danuthan wrote:
Hi,

I understand the approaches provided by experts.

However, in the real test with scrap paper, how can we draw a perfect circle?

The line in choice iii could have intersected the circle just a tiny bit.

Thanks,

In case of a doubt, just find the shortest distance of the line from the centre (0, 0). The circle with radius 5 will have every point at a distance of 5 from (0, 0). If the shortest distance of the line from (0, 0) is more than 5, it will not cut the circle at all.

The line intersects the x axis at 12 and y axis at -6. So it will form a right triangle with the axis such that hypotenuse is
$$\sqrt{12^2 + 6^2} = \sqrt{180} = 6*\sqrt{5}$$

Area of the triangle = (1/2)*Leg1*Leg2 = (1/2)*Altitude * Hypotenuse

$$6*12 = Altitude * 6*\sqrt{5}$$

$$Altitude = 2.4*\sqrt{5} = 2.4*2.2 = 5.3 (approx)$$

So shortest distance of the line from (0, 0) is 5.3 which is greater than 5. The line doesn't intersect the circle.

Here is the visual representation of Karishma's explanation. I just had the desire to draw the explanation and post it here.
Attachments

geo_8_sec_6_graphik_3.png [ 17.38 KiB | Viewed 679 times ]

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Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

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03 Aug 2018, 12:50
In this case any line which Y-intersect is -5 < x < 5 will intersect the circle in 2 points, then I and II is validated, we can hid of option A, B and D.
We just need to check III, here we depend on knowing how steep is the slope. As X is a variable then the line could intersect the circle in 2 points OR NOT. III is invalidated and we can eliminate option E.

Option C is the correct answer.
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Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

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15 Aug 2019, 15:13
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Re: A circle with center O and radius 5 is shown in the xy-plane   [#permalink] 15 Aug 2019, 15:13

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