M24 #12 - formatting error

Do lines y = ax^2 + b and y = cx^2 + d cross?

1. a = -c
2. b \gt d


Source: GMAT Club Tests - hardest GMAT questions

What am I supposed to do w/ #2?

I'd like to comment on this one:

First of all: equations given ARE NOT linear equations. We know that if two different lines do not cross each other they are parallel. How can we tell, based on the equations, whether the lines are parallel? We can check the slopes of these lines: parallel lines will have the same slopes. NOT that the slopes of lines must be negative reciprocals of each other (as it was mentioned in the earlier posts): in this case they are perpendicular not parallel.

Second of all: we have quadratic equations. These equations when drawn give parabolas. The question is: do they cross? This CAN NOT be transformed to the question: "are they parallel?" In the wast majority of cases the word "parallel" is used for the lines. Well we can say that concentric circles are parallel, BUT GMAT, as far as I know, uses this word ONLY about the lines (tutors may correct me if I'm wrong). Next, the word "parallel" when used for curves (lines, ...) means that these curves remain a constant distance apart. So strictly speaking two parabolas to be parallel they need not only not to intersect but also to remain constant distance apart. In this case, I must say that this can not happen. If a curve is parallel (as we defined) to the parabola it won't be quadratic: so curve parallel to a parabola is not a parabola. So I think that at this point we can stop considering this concept in regard to the original question.

So in which cases parabolas do not cross? There are number of possibilities: We can shift the vertex: the parabolas y=x^2 and y=x^2+1 will never intersect (note that they won't be exactly parallel but they will never intersect). We can consider downward and upward parabolas and in some cases they also never intersect... Of course there can be other cases as well.

As for the solution. We can follow the way dzyubam proposed (and I think it's the fastest way, provided we can identify correct examples) and consider two cases. First case: y=-x^2+1 and y=x^2+0 (upward and downward parabolas), which satisfies both statements, and see that in this case answer is YES, they cross each other; and the second case: y=x^2+1 and y=-x^2+0 (also upward and downward parabolas), which also satisfies both statements, and see that in this case answer is NO, they do not cross each other. Two different answers to the question, hence not sufficient.

Answer: E.

We can solve the question algebraically as well:

Do lines y = ax^2 + b and y = cx^2 + d cross?

(1) a = -c --> y_1= ax^2 + b and y_2=-ax^2 + d, now if they cross then for some x, ax^2+b=-ax^2 + d should be true --> which means that equation 2ax^2+(b-d)=0 must have a solution, some real root(s), or in other words discriminant of this quadratic equation must be \geq0 --> d=0-8a(b-d)\geq0? --> d=-8a(b-d)\geq0? Now can we determine whether this is true? We know nothing about a, b, and d, hence no. Not sufficient.

(2) b>d --> the same steps: if y_1= ax^2 + b and y_2= cx^2 + d cross then for some x, ax^2 +b=cx^2+d should be true --> which means that equation (a-c)x^2+(b-d)=0 must have a solution or in other words discriminant of this quadratic equation must be \geq0 --> d=0-4(a-c)(b-d)\geq0? --> d=-4(a-c)(b-d)\geq0? Now can we determine whether this is true? We know that b-d>0 but what about a-c? Hence no. Not sufficient.

(1)+(2) a=-c and b>d --> y_1= ax^2 + b and y_2=-ax^2 + d --> same steps as above --> 2ax^2+(b-d)=0 --> and the same question remains: is d=-8a(b-d)\geq0 true? b-d>0 but what about a? Not sufficient.

Answer: E.

Hope it helps.

You guys might have forgotten that we are dealing with quadratic equations here. The upward and downward parabolas may not cross even if the "slope" is different for the two equations. See the OE for more info.

The right answer is E.

This question is a bit different than a normal line equation. A normal line equation is y = [slope]x + [y intercept], but here the X is actually squared. This will lead to a curve similar to a parabola rather than a straight light. I don't think this is avery good question, but that's just me. I have not seen this before becuase I only did the first 12 or 13 math tests before my GMAT.

As for your questions prior to the DS question. Yes, generally, all you need to know for lines that cross is the slope. Remember that a Cartesian plane extends to infinity, so if the slopes are not IDENTICAL, then the lines will cross at some point. The y-intercept doesn't matter because at some point, the lines will cross unless they are EXACTLY parallel. To determine perpendicular lines, you are correct, the opposite sign (positive or negative) and reciprocal will be the test for perpendicular.

If slope is -\frac{1}{C} then the perpendicular slope will be C because we take the reciprocal so \frac{C}{1} and it's positive because the other was negative.

The OA is E.

Do the lines cross is another way of asking if they are parallel.

Statement 1 is sufficient. We know that if a = 1 and -c = a, then c = -1. These are not parallel. Also keep in mind the forumula for a line. Is y = [slope]x + [y-intercept] (or minus y-intercept).

Statement 2 is telling us information about the y-intercept. It is saying that no matter what d is, b is greater than d. This doesn't tell us any information about a or c, which are the most important because these two variables determine the slope of the line. Without knowing the slope of the lines, we cannot know if the lines cross.

Answer is A, and the anwer with the info in A is "Yes, the lines do cross."

I posted this during the rendering engine problem and (2) showed up as bd instead of b>d.

Thanks for the help, though.

what if a and c are both 0??? then we can't confidently say that the lines will cross. I think it should be E.

A

We can rewrite the question as 'are two lines y=ax^2+b and y=cx^2+d NOT parralel?'
They can be parralel only if have the same slope.

1) Slopes are different, so lines must cross, Suff.
2) there is no info about slopes, Insuff.

ssss

i get C..

one is an inverted parabola and we just need to make sure they dont have the same y intercept..

How about if a = -c = 0?

Answer should be E. We know (given S1) that one of them is upward parabola, the other is inverted. However, we don't know which one is inverted and which one is upward. I.e. - we dont know the sign of "a", it can be a=2 or a=-2

Hello, would someone pls answer my question here, thank you.

I believe it was several times in past test answers that you only need slopes of lines to determine whether or not they cross, the angle they form together or whether or not they are perpendicular. Assuming that S1 stated "a=-1/c" would we then know that two lines cross are perpendicular and therefore cross?
_________________________________
Do lines ax^2 + b and cx^2 + d cross?

1) a= -c
2) b > d

Answer: E

Would someone pls explain this question's answer further: What do we really need to know if two lines cross? The answer just gives examples. Also, if we have both slopes and know that one is the negative reciprocal of the other, do we then know they are perpendicular? Thankyou

Can some one explain this parabola and how to figure these out similar to normal lines?

I will feel good if I see such Q on the GMAT

Both statements required to answer this question.

Explanation:
Lets equate these two equations first.
We have no idea about the signs of a,b,c or d. so we cant say anything about the nature of the graph whether it is concave or convex.
After equating these equation we can get x^2 = (d-b)/(a-c).
For getting real values of x, the above result must satisfy that a!=c!=0.
Also, (d-b)/(a-c) must always be positive.
To find this we must have both the conditions. i.e, if a=-c and b>d then x^2 will be equal to a negative value so we can't have real values of x. Thus we can say that these two lines are not crossing each other.

How did you derive x^2= (d-b)/(a-c)?

Hi icandy

If we equate the two given equations, we get x^2=(d-b)/(a-c).
We need to equate these two equations, because if they cross each other, they must satisfy(must have equal values of y) for a specific value of x. we get this value of x from x^2=(d-b)/(a-c).

Or try this way:
y=ax^2+b
and
y=cx^2+d

equating these equations will give:
ax^2+b = cx^2+d
ax^2 - cx^2=d-b
x^2(a-c)=(d-b)
x^2=(d-b)/(a-c)

Hope this will help.

What is the OA for this question?

1. a = -c
2. b > d

1) If a = -c then first equation will become as follow,

1) -cx2 –y + b = 0
2) Cx2 –y + b = 0
As per rule two lines are parallel (if a1/a2 = b1/b2) but here a1/a2 = -1 and b1/b2 = 1 and so they are not equal and so line must intercept.

2) b > d, does not give enough information about a and c and so it is not sufficient.

Answer is A.