Lines that intersect the circle in 2 points

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Sun May 18, 2014 10:46 pm

Hello,

Can you please tell me how to solve this:

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

OA: C

I am thinking that in I, the y-intercept is 1 and the slope is -1. However, I don't know how to tell from this whether I intersects the circle in 2 points. Thanks a lot.

- Sri

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Source: Beat The GMAT — Problem Solving | Sun May 18, 2014 10:46 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon May 19, 2014 3:25 am

gmattesttaker2 wrote:Hello,

Can you please tell me how to solve this:

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

OA: C

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).

The correct answer is C.

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