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

Co-ordinate geometry is best done by making diagrams of your own. On the x axis, draw a circle with radius 5 and center at (0, 0). It will cut the x axis at 5 and -5 and y axis at 5 and -5.

Now try to plot the 3 given lines.

I. y = -x +1
Put x = 0, you get y = 1. So this line cuts the y axis at 1.
Put y = 0, you get x = 1. So this line cuts the x axis at 1.
This line, when extended on both sides will cut the circle at two distinct points.

II. y = 2x + 1
Similarly, plot this line and you will see that it will cut the circle at two points too.

III. y = (1/2)x - 6
This line cuts the y axis at -6 and x axis at 12. It is outside the circle and hence doesn't cut the circle.

Hence correct answer is (C)
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Check the diagram below:Answer: C.
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Find the intercepts by making once x=0 and then y=0. On finding these intercepts, draw line joining these points.
ex for line y=-x+1, the points will be (1,0) and (0,1). On joining these two points, it gets clear that the line will intersect the circle at two points. Similarly for II and III.
You will see that III is far away from the circle.
Hence I and II cut the circle.
+1C

If a line is intersecting a circle at 2 places, the distance between that line and the center of the circle< The radius

Also, we know that the distance between a point (p,q) and the line ax+by+c=0 is \(\frac{|ap+qb+c|}{\sqrt{a^2+b^2}\)

It's very easy to realise that (p,q) = (0,0) and the distance between the lines in I. and II. are less than 5. For III. the distance =\(\frac{|6|}{\sqrt{\frac{5}{4}}\) = \(\frac{12}{\sqrt{5}}\)>5

Thus, line in option III. doesn't intersect the circle at all.

C
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Hi Bunuel, sorry I did not get this around in my head... Could you explain in a bit more detail please?

I am bad with coordinate geometry, these things just dont seem to get into my head

i did this a much faster way (hopefully it works in all aspects).

I mad x 0 for every equation to find the y intercept.

I. Y = 1
II y = 1
III Y - -6

The circle only spans from -5 to 5 so only I and 2 will intersect in 2 points.

I understand the explanation, but not the question being ask by GMAT. " Lines that intersect the circle in 2 points include which of the following ?" is this question asking us identify the lines that intersect the circumference of the circle twice? Or the question is asking us to identify the lines that has two points within the circumference of the circle?

A line that intersects a circle in 2 points is one which has 2 points lying on the circumference of the circle (so it intersects the circumference twice i.e. to say it is not a tangent).
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Hi Karishma,
What if the first line,y = -x +1, has a restriction that x and y< +-5. Is it still going to qualify as a one of the sufficient answer choices?

Why do we assume the center is 0,0? I understand that the radius is 5 so it makes logical sense that the center is 0,0 but couldn't the center easily be for example (0,-2) and then the top point of the circle would be (0,3) and bottom be (0,-7). Radius is still 5 but the circle just sits in a different place in the coordinate plane.

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Hi healthjunkie,

The original prompt includes a picture that places O at the Origin. IF that drawing was NOT included with the question, and it wasn't clear that the circle was centered at the Origin, then your concerns would be valid. Having the picture to work with, are you comfortable answering the question?

GMAT assassins aren't born, they're made,
Rich

I am not sure I understand your question.
y = -x + 1 is the equation of a line (which is infinite on both ends) and it intersects the given circle at two points (4, -3) and (-3, 4). Absolute values of both x and y co-ordinates are less than |5| (if that's what you meant).
The line intersects the circle in two points and hence will be a part of the answer.
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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.
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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).


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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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Hey

So whenever center O is mentioned, we have to assume that center O is origin?

Gagan

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