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by danjuma » Fri Nov 26, 2010 11:16 pm
Circle C and line K lie in the XY plane . If circle C is centered at the origin and has radius 1, does line K intersect circle C?

1.The X-intercept of line K is greater than 1

2.The slope of line K is -1/10
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Source: — Data Sufficiency |
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by Night reader » Sat Nov 27, 2010 12:49 am
danjuma wrote:Circle C and line K lie in the XY plane . If circle C is centered at the origin and has radius 1, does line K intersect circle C?

1.The X-intercept of line K is greater than 1

2.The slope of line K is -1/10
IOM Answer C.

To determine whether line K intersects circle C we need the equation of the line for K and general equation for circle (x-a)^2+(y-b)^2=R^2 [a,b=0 origin, R=1). To find the equation, we need either two points, or one point and the slope. Both statements, on their own, are insufficient, as they provides only one point and the slope.

Taken together, two statements are sufficient: with one point, and the slope we can find the equation of line k. With the equation of the line K, we can answer the question.

y=ax+b, st(1) y=a*(x>1, say 11/10) + b, x intercept condition y=0 ____ 0=a*(x>1) + b
st(2) y=-1/10*x + b
st(1&2) y=-1/10*(x>1, x=11/10) + b; x intercept condition y=0 _____0=-1/10*(x>1, x=11/10) + b, b=11/10 (y intercept) because x=y E {-1;1} b=11/10 means that line K does not intersect the circle C. To test - plug in R {0;1 and 1;0} into line equation.
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by goyalsau » Sat Nov 27, 2010 4:47 am
I am not able to understand Night Reader How you assume that value of x and then you say y = x,

Please explain
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by Rahul@gurome » Sat Nov 27, 2010 6:12 am
While answering coordinate geometry problems (specially conceptual ones), drawing the scenario is the best method to solve the problem. Drawing helps you to visualize the problem and provides necessary ideas to make any assumptions. This problem for example can be solved by simply drawing the situation.

Given: Circle C and line K lie in the XY plane . Circle C is centered at the origin and has radius 1. To determine whether K intersects C or not we need to know the position of the line with respect to the circle.

Image
Refer to the above image. In the image line A and line B are parallel lines with slope -1/10. Say, x-intercept and y-intercept of line A are 20 an 2 respectively and those for line B are 0.5 and 5 respectively.

Statement 1: The X-intercept of line K is greater than 1.
If you have doubt about insufficiency of this statement, refer to the above image. Both line A and B have x-intercept greater than 1 (As the circle has radius 1, it crosses x-axis at (1,0)). But line A doesn't intersects the circle where line B does.

Not sufficient.

Statement 2: The slope of line K is -1/10.
Same as above! Line A doesn't, B does.

1 & 2 Together: Both lines have x-intercept greater than 1 and slope equal to -1/10. But one of them intersects and other doesn't.

Not sufficient.

The correct answer is E.
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