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Intern
Joined: 26 Jan 2006
Posts: 3

Do lines y = a*x^2 + b and y = c*x^2 + d cross each other ?
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23 May 2007, 18:25
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Do lines y = a*x^2 + b and y = c*x^2 + d cross each other ?
1) a = c
2) b > d
Can someone add details to the explanation below.. Thanks
Explanation
S1 + S2 is not sufficient. Consider y = x^2 + 1, y = x^2 + 0 (the answer is YES) and y = x^2 + 1, y = x^2 + 0 (the answer is NO). == Message from the GMAT Club Team == THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION. This discussion does not meet community quality standards. It has been retired. If you would like to discuss this question please repost it in the respective forum. Thank you! To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative  Verbal Please note  we may remove posts that do not follow our posting guidelines. Thank you.



Director
Joined: 26 Feb 2006
Posts: 823

Re: Do the lines cross each other ? DS Question
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23 May 2007, 19:02
kamal.gelya wrote: Do lines y = a*x^2 + b and y = c*x^2 + d cross each other ?
1) a = c 2) b > d
Can someone add details to the explanation below.. Thanks
Explanation
S1 + S2 is not sufficient. Consider y = x^2 + 1, y = x^2 + 0 (the answer is YES) and y = x^2 + 1, y = x^2 + 0 (the answer is NO).
imo,
from i, if slopes are not equal, then the lines cros each other.
from ii, if intercepts are not equal, they still have same slop and can be parallel.
So A.



SVP
Joined: 01 May 2006
Posts: 1740

(E) for me
We have 2 curves with so these caracteristics:
> a and c represent how much "narrow" the curve will be. For instance:
 if a is very negative (100, 10000), the montain created by the curve will be very narrow (even a pic).
 if a is very positive (100, 10000), the valley created by the curve will be very narrow (even a fail).
> b and d are the Yinterceptors and so gives the positions of the extremum of the 2 curves, maximum or minimum, depending on the sign of a and c.
From 1
a=c. Thus, the 2 curves represents a montain (the negative one from a and c) and a mirror of it, a valley.
o If the Y interceptor of the montain is equal to the one of the valley, there is 1 crossing point, their extremum
o If the Y interceptor of the montain is superior to the one of the valley, there are 2 crossing points.
o If the Y interceptor of the montain is inferior to the one of the valley, there is no crossing points.
INSUFF.
From 2
b > d. We have to know a relationship between a and c.
o If a > c, there is no crossing point.
o If 0 < a < c, there are 2 crossing points
INSUFF.
From 1 and 2
b > d and a=c
We have:
o If a > 0 > c, the valley a,b never crosses the montain c,d.
o If c > 0 > a, the montain a,b crosses the valley c,d at 2 points.
INSUFF.



Manager
Joined: 30 Mar 2007
Posts: 185

fig ,
We have 2 curves with so these caracteristics:
> a and c represent how much "narrow" the curve will be. For instance:
 if a is very negative (100, 10000), the montain created by the curve will be very narrow (even a pic).
 if a is very positive (100, 10000), the valley created by the curve will be very narrow (even a fail).
i am not able to understand your point.
y = a*x^2 + b
x=+sqrt((yb)/a);
Is in not that as a increases curve will have lesser width.



GMAT Instructor
Joined: 04 Jul 2006
Posts: 1243
Location: Madrid

Re: Do the lines cross each other ? DS Question
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24 May 2007, 02:48
kamal.gelya wrote: Do lines y = a*x^2 + b and y = c*x^2 + d cross each other ?
1) a = c 2) b > d
Can someone add details to the explanation below.. Thanks
Explanation
S1 + S2 is not sufficient. Consider y = x^2 + 1, y = x^2 + 0 (the answer is YES) and y = x^2 + 1, y = x^2 + 0 (the answer is NO).
It is deceiving to call these lines, by the way!
They will cross if and only if (ac)x^2= db has more than one solution i.e if (db)/(ac) > 0. (2) tells us that the numerator is >0, but if a= c as per (1), the denominator can be positive or negative. (1) and (2) together are insufficient



SVP
Joined: 01 May 2006
Posts: 1740

apache wrote: fig , We have 2 curves with so these caracteristics: > a and c represent how much "narrow" the curve will be. For instance:  if a is very negative (100, 10000), the montain created by the curve will be very narrow (even a pic).  if a is very positive (100, 10000), the valley created by the curve will be very narrow (even a fail).
i am not able to understand your point.
y = a*x^2 + b
x=+sqrt((yb)/a);
Is in not that as a increases curve will have lesser width.
Yes.... The absolute value of a gives the width ... I mean a very high increase, a = a +100000 (a montain) or a very high decrease, a = a 100000 (a valley) both can decrease the width and oddly mirror the montain to a valley or the valley to a montain



Manager
Joined: 04 May 2007
Posts: 110

hrmm. this question seems odd, where did it come from? i thought lines had to be straight? i said A for the same reason as Himalayan.... if they aren't parallel they will intersect at some point in space.



Director
Joined: 26 Feb 2006
Posts: 823

Fig wrote: (E) for me We have 2 curves with so these caracteristics: > a and c represent how much "narrow" the curve will be. For instance:  if a is very negative (100, 10000), the montain created by the curve will be very narrow (even a pic).  if a is very positive (100, 10000), the valley created by the curve will be very narrow (even a fail). > b and d are the Yinterceptors and so gives the positions of the extremum of the 2 curves, maximum or minimum, depending on the sign of a and c. From 1a=c. Thus, the 2 curves represents a montain (the negative one from a and c) and a mirror of it, a valley. o If the Y interceptor of the montain is equal to the one of the valley, there is 1 crossing point, their extremum o If the Y interceptor of the montain is superior to the one of the valley, there are 2 crossing points. o If the Y interceptor of the montain is inferior to the one of the valley, there is no crossing points. INSUFF. From 2b > d. We have to know a relationship between a and c. o If a > c, there is no crossing point. o If 0 < a <c> d and a=c We have: o If a > 0 > c, the valley a,b never crosses the montain c,d. o If c > 0 > a, the montain a,b crosses the valley c,d at 2 points. INSUFF.
I could not understant what you and Kevincan are telling?
Can you guys explain by figure? thanks.



SVP
Joined: 01 May 2006
Posts: 1740

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Updated on: 26 May 2007, 01:06
Ok... I can try

From 1
a=c. Thus, the 2 curves represents a montain (the negative one from a and c) and a mirror of it, a valley.
o If the Y interceptor of the montain is equal to the one of the valley, there is 1 crossing point, their extremum CASE A
o If the Y interceptor of the montain is superior to the one of the valley, there are 2 crossing points. CASE B
o If the Y interceptor of the montain is inferior to the one of the valley, there is no crossing points. CASE C
INSUFF.

CASE A: Arbitrarly, I take a=1 and so c=1. This way, a,b will be the valey and c,d the montain. I choose also b=d=1. (Fig A)
CASE B: Arbitrarly, I take a=1 and so c=1. Then, we have b < d for this case and so I take b=1 and d=2. (Fig B)
CASE C: Arbitrarly, I take a=1 and so c=1. Then, we have b > d for this case and so I take b=2 and d=1. (Fig C)
Attachments
Fig A_a equal 1_c equal 1_b & d equal 1.gif [ 3.95 KiB  Viewed 1818 times ]
Fig B_a equal 1_c equal 1_b equal 1_d equal 2.gif [ 4.01 KiB  Viewed 1818 times ]
Fig C_a equal 1_c equal 1_b equal 2_d equal 1.gif [ 3.85 KiB  Viewed 1817 times ]
Originally posted by Fig on 26 May 2007, 00:46.
Last edited by Fig on 26 May 2007, 01:06, edited 2 times in total.



SVP
Joined: 01 May 2006
Posts: 1740

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Updated on: 26 May 2007, 01:06
From 2
b > d. We have to know a relationship between a and c.
o If a > c, there is no crossing point. CASE D
o If 0 < a < c, there are 2 crossing points CASE E
INSUFF.

CASE D: I choose a=2 and c=1 as well as b=2 and d=1 (Fig D)
CASE E: I choose a=1 and c=2 as well as b=2 and d=1 (Fig E)
Attachments
Fig D_a equal 2_c equal 1_b equal 2_d equal 1.gif [ 3.73 KiB  Viewed 1814 times ]
Fig E_a equal 1_c equal 2_b equal 2_d equal 1.gif [ 3.71 KiB  Viewed 1816 times ]
Originally posted by Fig on 26 May 2007, 00:55.
Last edited by Fig on 26 May 2007, 01:06, edited 2 times in total.



SVP
Joined: 01 May 2006
Posts: 1740

From 1 and 2
b > d and a=c
We have:
o If a > 0 > c, the valley a,b never crosses the montain c,d. CASE F
o If c > 0 > a, the montain a,b crosses the valley c,d at 2 points. CASE G
INSUFF.

CASE F: I choose b=2 and d=1 as well as a=1 and c=1. (Fig F)
CASE G: I choose b=2 and d=1 as well as a=1 and c=1. (Fig G)
Attachments
Fig F_a equal 1_c equal 1_b equal 2_d equal 1.gif [ 3.78 KiB  Viewed 1813 times ]
Fig G_a equal 1_c equal 1_b equal 2_d equal 1.gif [ 3.96 KiB  Viewed 1814 times ]



Senior Manager
Joined: 18 Jul 2006
Posts: 468

gmatiscoming wrote: hrmm. this question seems odd, where did it come from? i thought lines had to be straight? i said A for the same reason as Himalayan.... if they aren't parallel they will intersect at some point in space.
These are parabola, not straight lines.



Manager
Joined: 04 May 2007
Posts: 110

juaz,
i know they are parabola. i just answered the Question as A because I thought lines had to be straight, therefore they would intersect if not parallel... the question did not say they were curves. it was a time saving method... ya know, don't do the whole problem if you don't have to kinda thing..
i am curious if this is an official question... if someone could point me to the definition of what a line is within the context of coordinate geometry i would be thankful; i want to know what other quirks 'lines' might have so i don't make a similar mistake again.



CEO
Joined: 15 Aug 2003
Posts: 3339

Dear all
I think this question is from one of the early challenges. This question is supposed to be for straight lines, not a curve. We apologize for this mistake.
The correct version should read:
Do lines $$y = a*x_1 + b$$ and $$y = c*x_2 + d$$ cross each other ?
1) a = c
2) b > d
The specific property we were trying to test was the following:
Two lines are perpendicular if their respective slopes are negative reciprocals of one another. Thus, the line $$ y= a * x_1 +b $$ would be perpendicular to any line whose slope is $$ 1/a $$ .



Manager
Joined: 04 May 2007
Posts: 110

praetorian,
so can i stop wetting my pants and trembling in the fetal posistion in a corner now at the thought of having to relearn what a line can and cant be? lines *are* straight on the GMAT? right?
::right??::



CEO
Joined: 15 Aug 2003
Posts: 3339

gmatiscoming wrote: praetorian,
so can i stop wetting my pants and trembling in the fetal posistion in a corner now at the thought of having to relearn what a line can and cant be? lines *are* straight on the GMAT? right?
::right??::
Yes, you may stop wetting your pants. Lines are straight. If you want to be more formal, horizontal and vertical are special cases of straight lines.
As a side note, thank you for taking the problem apart. == Message from the GMAT Club Team == THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION. This discussion does not meet community quality standards. It has been retired. If you would like to discuss this question please repost it in the respective forum. Thank you! To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative  Verbal Please note  we may remove posts that do not follow our posting guidelines. Thank you.



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Re: Do lines y = a*x^2 + b and y = c*x^2 + d cross each other ?
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31 Mar 2019, 21:51
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Re: Do lines y = a*x^2 + b and y = c*x^2 + d cross each other ?
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