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DS Does the equation y = ( x – p)( x – q) intercept the

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Source: — Data Sufficiency |
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by OptimusPrep » Wed Aug 05, 2015 9:01 pm
For a point to lie on the lie, the X, y co ordinates of the point should satisfy the equation of the line

So, 0 = (2-p)(2-q)
On solving, we get

pq + 4 -2(p + q) = 0

If we are able to prove that LHS is indeed 0, then the line passes through the point.
So our question remains:

Is pq + 4 -2(p + q) = 0?

Statement 1:
This tells us about pq, but nothing about (p + q). Hence not sufficient

Statement 2:
This tells us about (p + q) but nothing about pq. Hence not sufficient

Combining the two gives us all that we need.
On putting the values, we can see that the term pq + 4 -2(p + q) is indeed = 0

Hence C is the answer
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by [email protected] » Wed Aug 05, 2015 10:01 pm
Hi rommysingh,

This is a 'quirky' graphing question that involves a relatively rare issue - the formula involved is for a parabola (although you don't need to know that to answer the question).

We're asked if (2,0) is one of the points that 'fits' the equation Y = (X-P)(X-Q). This is a YES/NO question. Before dealing with the 2 Facts, we can plug in those values for X and Y....then the question becomes....

Does 0 = (2-P)(2-Q)?

Looking at the question in this way, it's really asking...."Does P=2 and/or does Q=2?"

Fact 1: (P)(Q) = -8

IF....
P=1
Q=-8
Then the answer to the question is NO.

IF...
P=2
Q=-4
Then the answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: -2 - P = Q

We can rewrite this as...
P + Q = -2

IF....
P = -1
Q = -1
Then the answer to the question is NO

IF...
P = 2
Q = -4
Then the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, we know...
(P)(Q) = -8
P + Q = -2

This is a 'system' of equations, which means that we CAN solve for P and Q (one of them would be -4 and one would be +2). While we don't know which one would be which, since one of them MUST be +2, the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

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Last edited by [email protected] on Thu Aug 06, 2015 2:19 pm, edited 1 time in total.
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by GMATGuruNY » Thu Aug 06, 2015 3:04 am

This is a 'quirky' graphing question that involves a relatively rare issue - the formula involved is for a circle.
y = (x-p)(x-q) is a formula not for a circle but for a PARABOLA.
In the equation for a circle, both x and y are squared.
Here, only the x-value is squared:
y = (x-p)(x-q)
y = x² - px - qx + pq
y = x² - (p+q)x + pq.
Combined, we know...
(P)(Q) = -8
P + Q = -2

This is a 'system' of equations, which means that we CAN solve for P and Q
A system of two LINEAR equations with two variables can always be solved, but the equation in statement 1 is not linear.
A linear equation can be written in the following form:
y = mx + b.
The equation in statement 1 cannot be written in this form.

Given two variables, one linear equation, and one non-linear equation, we might not have sufficient information to solve.
Consider the following revision of the posted problem:
Does the equation y = (x - p)(x - q) intercept the x-axis at the point (2,0)?

(1) pq = 8

(2) 2 = p-q
Question stem, rephrased:
Does p=2 or q=2?

Here, the two statements combined yield the following cases:
Case 1: p=4, q=2
In this case, the answer to the question stem is YES.
Case 2: p=-2, q=-4
In this case, the answer to the question stem is NO.
Thus, in this revision of the posted problem, the two statements combined are INSUFFICIENT.
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by [email protected] » Thu Aug 06, 2015 2:19 pm
Hi rommsingh,

Mitch correctly pointed out that the equation is for a parabola (and not a circle). I've edited by post to correct that error. However, he truncated what I posted about the two equations. If you read the full paragraph, you'll notice how I stated how one of the variables would be -4 and the other would be +2, BUT we wouldn't know which one was which. However, since the question 'hinged' on EITHER of them equaling +2, then it wouldn't matter.

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by rommysingh » Thu Aug 06, 2015 6:48 pm
thanks!
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by GMATGuruNY » Fri Aug 07, 2015 5:02 am
[email protected] wrote:Hi rommsingh,

Mitch correctly pointed out that the equation is for a parabola (and not a circle). I've edited by post to correct that error. However, he truncated what I posted about the two equations. If you read the full paragraph, you'll notice how I stated how one of the variables would be -4 and the other would be +2, BUT we wouldn't know which one was which. However, since the question 'hinged' on EITHER of them equaling +2, then it wouldn't matter.

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Rich, your solution is great.
With regard to the portion I quoted, my concern was that students would overgeneralize, concluding that any DS with two variables, one linear equation and one non-linear equation can be solved.
Here, the two equations are sufficient only because the values of p and q are interchangeable.
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