A Parabola in the Coordinate Geometry Plane is Represented GMAT Data sufficiency

bySayantani Barman Experta en el extranjero

Question: A parabola in the coordinate geometry plane is represented by the equation y = x^2 + k, where k is a constant greater than 0. Line L intersects this parabola at exactly one point. Is this point of intersection in Quadrant I?

(1) The slope of line L is positive.
(2) x is greater than 0.

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D
Solution and Explanation:
Approach Solution 1:

You are informed that the parabola's equation is
Where k is POSITIVE (let's suppose k = 2), y = x2 + k. If so, y = x2 + 2.

Since x2's coefficient is positive, the parabola will be upward-looking.
Since x2 cannot be negative, x2 + 2 has no real roots because it can never equal 0. Consequently, it won't cut the x-axis anywhere.

The parabola cuts the y-axis at 0 when x = 0 and y = 2. (0, 2).
The parabola will be symmetric about the y-axis because there is no bx term (for both x = 3 and -3, y will be 11 etc.).
The parabola will therefore be symmetric about the y axis, above the x axis, and intersect the y axis at point k. (which we assumed 2 for ease).
Therefore, case 2 of your illustration is illogical. A parabola MUST appear as in example 1.

D is the correct answer.

Approach Solution 2:

You can see that this parabola opens upward, is centered on the y-axis, and has its lowest point (vertex) above the x-axis (since k is positive) if you have a rudimentary understanding of curved lines in the coordinate geometry plane. Make sure to take the time to quickly sketch down this parabola before you begin to analyze each statement because it is crucial to this problem that you have a proper visual representation of it.

Assuming line L precisely meets this parabola at one point, if you've drawn the parabola correctly, you can visually observe various possibilities along the parabola. The parabola does not even touch the other two quadrants, making the only junction points in Quadrants I and II obvious. Whether or not that junction point is in quadrant I is the subject of the inquiry.

You can infer from Statement 1 that Line L has a positive slope. You can show that no matter the line, it must intersect the parabola in the first quadrant if you draw a few lines with positive slopes that only cross the curve once. The y axis would be the point of intersection if the slope were 0 (which is not permitted), and as the slope increases from 0 to infinity, it can only intersect in the first quadrant. The first statement is adequate. Take away (B), (C), and (E).

You can only find intersection sites in the first quadrant if x is positive, as all x values in the second quadrant are negative, according to statement 2. Statement 2 is likewise adequate, and the correct response is (D).

D is the correct answer.

Approach Solution 3:

First, here is a picture of the y = x2 graph:

All of the y-coordinates on the graph of y = x2 will increase by k units as we are adding k, which is positive.
Consequently, the graph of y = x2 + k will resemble this:

The point of intersection (of the line and parabola) must be in either quadrant I or II since the graph of y = x2 + k only lies in quadrants I and II.

Statement 1: Line L has a positive slope
This is all that has to be said. This is why:
A line with a positive slope will also cross the parabola at a location in quadrant I if it meets the parabola in quadrant II.
For instance:

We may be positive that line L cannot intersect the parabola in quadrant II because we are informed that line L intersects this parabola at exactly one spot.
Line L must therefore only cross the parabola in quadrant I.
Statement 1 is sufficient because we are confident in our ability to respond to the target question.

Here's another angle to consider if you're still not persuaded.
Line L must be tangent to the parabola if it crosses it precisely once along its length.
Line L, for instance, would look like this:

Or as follows:

As you can see, the point of intersection for line L must be in quadrant I if it has a positive slope and is tangent to the parabola.

Second claim: x is greater than 0.
I don't particularly like this phrasing.
I would phrase sentence 2 as follows for clarity: The junction point's x-coordinate is positive.
This indicates that line L must be perpendicular to the parabola at one of the below-shown red dots.

We may be positive that line L must cross the parabola in quadrant I because all of the intersection points are in quadrant I.
Statement 2 is sufficient since we are confident in our ability to respond to the target question.

Response: D

“A Parabola in the Coordinate Geometry Plane is Represented GMAT Data sufficiency” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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A Parabola in the Coordinate Geometry Plane is Represented GMAT Data sufficiency

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