In the xy-plane, does the line L intersect the graph of y = x^2 ?

In the XY-Plane does line l intersect the graph of y=x^2?

(1) Line l passes through the point (4, -8). Consider the diagram below:



As you can see line passing through (4, -8) may or may not intersect with the graph of y=x^2. Not sufficient.

(2) Line l passes through the point (-4, 16). Since (-4)^2=16, then point (-4, 16) is ON the graph of y=x^2, thus line passing through this point intersects the graph of y=x^2. Sufficient.

Answer: B.

Hope it's clear.


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1) This is clearly insufficient. It is possible that the line might intersect the curve at 4,16 ( if it has slope 0, like x=4), or might not intersect at all, like line y=-8.

2) This is sufficient as the line satisfies the equation of the curve y=x^2, 16=(-4)^2, therefore the line and curve intersect at this point.

Answer should be B.

Hi Gurus,

My question may be absurd but please help me understand the concept here. When I see y=x^2 i do not see it as an Upright parabola but a parabola drawn towards positive x axis . Say y=-3 then x^2 = 9 so (-3,9) is on the parabola, similarly (-2,4),(-1,1),(0,0),(1,1) and (2.4) all should form the parabola with function y=X^2.

Now if line l passes through (-4,16)- statement B, i see it as x co-ordinate as -4 and y co-ordinate as 16 then this point does not lie on the parabola. Can anyone please explain where am I at fault?

Thanks in advance.

Arun

I think you should brush-up fundamental on coordinate geometry:


As for your question, check the graph of y=x^2 and the point (-4, 16) on it:To check whether (-4, 16) is on y=x^2, substitute x=-4 there and check whether y comes out to be 16: (-4)^2=16, so (-4, 16) is on y=x^2.

Hope it helps.
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I understand the everything concerning how line l intersects y=x^2 at (-4,16) and not (4,-8), but I am confused as to what the question is asking. As a data sufficiency problem, are we not simply supposed to select the statements that allow us to reach an answer? For example, this question specifically asks whether line intersects y=x^2; then, is the first statement not sufficient NOT simply because (4,-8) isn't on y=x^2 but because even if it's not, we don't know if it intersects at another point (i.e. a vertical line at x=4 vs. a horizontal line at y=-8)? Just trying to make sure I'm not getting the right answer for the wrong reason.

Thanks in advance!

There are two kinds of data sufficient questions: YES/NO DS questions and DS questions which ask to find a value.

In Yes/No Data Sufficiency questions, statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no".

When a DS question asks about the value of some variable, then the statement is sufficient ONLY if you can get the single numerical value of this variable.


The original question is an Yes/No Data Sufficiency question. And the first statement is not sufficient because we can have both yes and no answers to the question. See my post with diagram in it.

Hope it helps.
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

In the xy-plane, does the line L intersect the graph of y = x^2

(1) Line L passes through (4, -8)
(2) Line L passes through (-4, 16)

There are 2 variables (slope, y-intercept) and 2 equations are given by the 2 conditions, so there is high chance (C) will be the answer.
C actually is the answer, but is too trivial.
If we look at the conditions separately,
condition 1 cannot determine anything
Condition 2 y=x^2 passes through (-4,16) so it's a 'yes' and is sufficient, making the answer (B).

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

VeritasKarishma Just wanted to clarify a concept am having some doubts about pertaining to this question. Statement two clearly gives a yes as it satisfies y=x^2. But as for statement 1 it doesn't satisfy it i.e. it doesn't lie on the graph y=x^2 but we can't say its a clear NO(i.e Sufficient) because if two lines intersect they need to have only 1 solution in common i.e the line MAY still intersect the parabola.

What am trying to clarify is that for two lines to intersect we just need a common solution and not for all points (in this case 4,-8) to satisfy both equations?

Hope am able to explain my query. Looking forward to hear from you.
Best

Sidharth003 - I am afraid I don't really understand your question.

Statement 1 is not sufficient because given one point, we know that a line can pass through it in infinite ways. Look at the diagram.


Some of these lines will intersect the parabola but the line m parallel to X axis will not intersect the parabola.

When two lines intersect, they have exactly one point common to both lines.
When two lines have all points common, they are overlapping lines i.e. they are the same line.
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Could you please explain why the approach towards both the points are different? ie; I agree with the first approach where we draw the parabola and draw the possibilities of how a line passing through point (4,-8) could be anything (parallel to x axis, or positive sloped, negative sloped.
But when we approach the second point, using the same principle, would that not mean that a line passing through (-4,16) can also look either parallel to x axis, positive sloped, negative sloped?

Be attentive to the question being asked, which is not about the slope of line L but rather whether it intersects the graph of y = x^2. In (1), line L intersects the graph of y = x^2 in some cases, but not in others. However, in (2), we know that line L passes through the point (-4, 16), which is already on the graph of y = x^2. Therefore, it follows that the line passing through this point always intersect the graph of y = x^2.
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