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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.

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.

Re: Do the lines cross each other ? DS Question [#permalink]

Show Tags

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 (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

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

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.

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?

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 1362 times ]

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)

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.

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.

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 $$ -1/a $$ .

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?

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.