GMAT Prep

moneyman wrote:I have a tough time with questions on co-ordinate geometry because I dont know what to look for in the question..pls help!!


This one is from GMAT Prep

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

We can draw the circle based on the info given, so we need information about line k.

(1) We know that k crosses the x-axis to the right of "1", but have no idea what the slope is: insufficient.

(2) We know the slope, but we have no idea where the line exists in the x-y plane: insufficient.

Together:

a slope of -(1/10) means that line k is very flat - it goes up 1 for every 10 it goes to the left.

So, if k passes through the point (2,0), it would definitely intersect the circle. However, if k passes through the point (10000,0), it's going to miss the circle by a mile.

So, even after combining statements (1) and (2), we're not sure if k intersects the circle: choose (e).

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a slope of -(1/10) means that line k is very flat - it goes up 1 for every 10 it goes to the left.

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Confused on how we can tell this line is flat. Guess its a very basic slope question but doesnt -1/10 slope mean down 1 and 10 to the right?

The following 2 sentences are equivalent

"it goes up 1 for every 10 it goes to the left."

"it goes down 1 and 10 to the right"

First sentence starts from the right of teh coordinate system and moves up the line to the left

Second sentence starts from the left of the coordinate system and moves down the line to the right.

It is the same line ... just depends which way you look at the line.
Hope this helps

Stuart Kovinsky wrote:

We can draw the circle based on the info given, so we need information about line k.

(1) We know that k crosses the x-axis to the right of "1", but have no idea what the slope is: insufficient.

(2) We know the slope, but we have no idea where the line exists in the x-y plane: insufficient.

Together:

a slope of -(1/10) means that line k is very flat - it goes up 1 for every 10 it goes to the left.

[/quote]

Is there any information we could make out just from the slope of a line? Especially if the slope were "positive" vs "negative". Would we even be able to tell what quadrant a line passes through if a slope is NEGATIVE or POSITIVE. Appreciate your response.

ildude02 wrote:Is there any information we could make out just from the slope of a line? Especially if the slope were "positive" vs "negative". Would we even be able to tell what quadrant a line passes through if a slope is NEGATIVE or POSITIVE. Appreciate your response.

Lines with a positive slope always pass through quadrants I and III. They also pass through exactly 1 of quadrant II, quadrant IV or the origin (i.e. 0,0).

Lines with a negative slope always pass through quadrants II and IV. They also pass through exactly 1 of quadrant I, quadrant III or the origin.

Stuart Kovinsky wrote:

ildude02 wrote:Is there any information we could make out just from the slope of a line? Especially if the slope were "positive" vs "negative". Would we even be able to tell what quadrant a line passes through if a slope is NEGATIVE or POSITIVE. Appreciate your response.

Lines with a positive slope always pass through quadrants I and III. They also pass through exactly 1 of quadrant II, quadrant IV or the origin (i.e. 0,0).

Lines with a negative slope always pass through quadrants II and IV. They also pass through exactly 1 of quadrant I, quadrant III or the origin.

Stuarts, thanks for your response. I'm wondering would this information ever be useful for solving any problem(I assume so, but not sure if there is any good example to test it. I tried to apply this to the above problem and it didn't help me much)?

I got the correct answer but am not too sure..
the slope is given at -1/10..what does this say?That the slope is -ive so passes through quad 2 and 4...

but what about the inclination?if its almost parallel to the x-axis it may intersect the circle..not so if the line is almost parallel to y axis..

Basically does the figure of 1/10 tell us anything?What if it was given -5?Would the answer be diff.?

Would like to get the answer without calculations:)Please help.

i ll try to do it
you know that every line might be expressed as y=kx+b, where k is a slope. Consider b=0 first.
we have an equation y=kx ( y=-1/10x, in particular)
Literally, slope tells us the distance in the y-axis per x, but affected by the slope.
Our equation tells us that, if x=1, y will travel to -1/10. if x=2, y will travel to -1/5.
For our problem we should consider x=1 and y=1, because the radius of a circle is 1.
Of coarse, if b=0 ==> y=-1/10x. Line obviously will intersect the circle.
Now it all depends on variable b. if b is 1000000, line won't intersect the circle. P.s. statements do not tell us, where exactly like k intersects x axis unfortunately. Hence E.

@Stuart, I am not sure I understand this. Isn't there only ONE AND ONLY ONE line with a slope -1/10? Or can we draw several lines with this slope?

Thouraya wrote:@Stuart, I am not sure I understand this. Isn't there only ONE AND ONLY ONE line with a slope -1/10? Or can we draw several lines with this slope?

Hi,

there's an infinite number of lines with that slope. For example:

y = (-1/10)x + 1
y = (-1/10)x + 2
y = (-1/10)x + 3

and so on....

Each of those lines is parallel, but crosses the y-axis in a different spot.

baladon99 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

One approach is to draw the circle and the line to see whether the two must intersect.

X-intercept is greater than 1, slope = -1/10, line k intersects circle C:



X-intercept is greater than 1, slope = -1/10, line k does not intersect circle C:



Thus, the 2 statements combined are insufficient to determine whether line k intersects circle C.

The correct answer is E.

what about if we had a different value for the slope, but still negative?
would that change anything?