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Do lines y = a*x^2 + b and y = c*x^2 + d cross each other ?

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Do lines y = a*x^2 + b and y = c*x^2 + d cross each other ? [#permalink] New post 23 May 2007, 17:25
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A
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D
E

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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).
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Re: Do the lines cross each other ? DS Question [#permalink] New post 23 May 2007, 18:02
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.
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 [#permalink] New post 23 May 2007, 23:01
(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 Y-interceptors 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.
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 [#permalink] New post 24 May 2007, 01:27
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((y-b)/a);

Is in not that as a increases curve will have lesser width.
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Re: Do the lines cross each other ? DS Question [#permalink] New post 24 May 2007, 01:48
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 (a-c)x^2= d-b has more than one solution i.e if (d-b)/(a-c) > 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
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 [#permalink] New post 24 May 2007, 03:41
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((y-b)/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 :)
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 [#permalink] New post 25 May 2007, 16:22
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.
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 [#permalink] New post 25 May 2007, 19:43
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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 Y-interceptors 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> 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.
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 [#permalink] New post 25 May 2007, 23:46
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
Fig A_a equal 1_c equal -1_b & d equal 1.gif [ 3.95 KiB | Viewed 1135 times ]

Fig B_a equal 1_c equal -1_b equal 1_d equal 2.gif
Fig B_a equal 1_c equal -1_b equal 1_d equal 2.gif [ 4.01 KiB | Viewed 1136 times ]

Fig C_a equal 1_c equal -1_b equal 2_d equal 1.gif
Fig C_a equal 1_c equal -1_b equal 2_d equal 1.gif [ 3.85 KiB | Viewed 1135 times ]


Last edited by Fig on 26 May 2007, 00:06, edited 2 times in total.
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 [#permalink] New post 25 May 2007, 23:55
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
Fig D_a equal 2_c equal 1_b equal 2_d equal 1.gif [ 3.73 KiB | Viewed 1132 times ]

Fig E_a equal 1_c equal 2_b equal 2_d equal 1.gif
Fig E_a equal 1_c equal 2_b equal 2_d equal 1.gif [ 3.71 KiB | Viewed 1133 times ]


Last edited by Fig on 26 May 2007, 00:06, edited 2 times in total.
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 [#permalink] New post 26 May 2007, 00:03
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
Fig F_a equal 1_c equal -1_b equal 2_d equal 1.gif [ 3.78 KiB | Viewed 1129 times ]

Fig G_a equal -1_c equal 1_b equal 2_d equal 1.gif
Fig G_a equal -1_c equal 1_b equal 2_d equal 1.gif [ 3.96 KiB | Viewed 1129 times ]

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 [#permalink] New post 26 May 2007, 00:31
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.
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 [#permalink] New post 26 May 2007, 07:52
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.
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 [#permalink] New post 26 May 2007, 08:09
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 $$ -1/a $$ .
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 [#permalink] New post 26 May 2007, 09:32
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??::
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 [#permalink] New post 26 May 2007, 09:43
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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. :twisted:

As a side note, thank you for taking the problem apart.
  [#permalink] 26 May 2007, 09:43
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