A parabola in the coordinate geometry plane is represented by the equa GMAT data sufficiency

Question: A parabola in the coordinate geometry plane is represented by the equation y = x² + 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?

Statement (1) The slope of line L is positive.
Statement (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.

Answer: D

Solution and Explanation:

Approach Solution 1:
Given: The equation y = x2 + k, where k is a constant greater than 0, represents a parabola in the coordinate geometry plane. This parabola is precisely intersected by line L at one point.
Does the junction point fall within Quadrant I?
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.
Hence, 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.
Assertion 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:

Since we know that there is only one point where line L crosses the parabola in quadrant II, we can be confident that it does not do so at any other point.
That means the parabola can only be intersected by Line L in Quadrant I.
Confidence in our abilities to answer the inquiry at hand leads us to conclude that Statement 1 is adequate.
If you're still not convinced, maybe this will help.
If line L crosses the parabola exactly once along its length, then it is tangent to the parabola.
For example, here is how Line L would appear on a map:

Or like this

If line L has a positive slope and is tangent to the parabola, then its point of intersection must be in quadrant I.
The second hypothesis is that x > 0.
This is wording that I don't find appealing.
To be more precise, I suggest rephrasing the second statement as follows: The x-coordinate of the intersection is in the positive range.
This implies that one of the red dots displayed below must be perpendicular to line L, where the parabola and line are both represented.

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.
Correct option: D

Approach Solution 2:
Provided is the quadratic equation y=x²+k, where k>0.
The QE's nature is:
- a) Facing up (a > 0), above the x-axis (c = k > 0), and symmetric about the y-axis (x = -b/2a = 0).
a line and a parabola cross:
- b) Absence of tangents to outside points:
2, where one tangent line has a positive slope and the other a negative slope.
The line y=k with no slope is the third tangent to the parabola.
d) The parabola is cut by the line x=0 at a single point.
slope=infinity
e) When the slope of the line is solely positive, the point of intersection of the line and parabola remains at Q1.
Note that a tangent can also be an intersection.
Assuming that Line L crosses this parabola at only one point, it follows that either the tangent with a positive slope, negative slope, or zero slope should be taken into consideration.
St. 1: The slope of line L is positive. Please refer to point (e).
x is bigger than 0 in St2.
Y is positive since k is positive. As a result, the quadratic equations x and y are both positive.
As a result, this section shows the positive symmetry of the y-axis.
implies that line L has a positive slope.
(Because the right side of the parabola's y-axis must have precisely one intersection point)
Refer to point (e), that's all.
Correct option: D

Approach Solution 3:
We can claim that the following situations can lead to intersection points:
On the y-axis, row
2) About Q1
3) A vertical line on the x-axis that precisely intersects the parabola at one location
The x-axis.
Analyzing each of the aforementioned situations
1) With x=0 and the y-axis, There is a warning though, as stated in statement 2, "x" is only positive. Discard. N.B:- You are aware that 0 is not positive or negative.
We are unable to take this case into consideration.
2) On the first question: you answered it right, and it is evident.
3) A vertical line traveling through the x-axis and intersecting the parabola on Q1
When provided, y=x²+k
As stated, k>0 As per statement 2, x>0 Hence, y > 0 As x > 0, y > 0, the intersection point is in Q1.
4) On the x-axis: The parabola is above the x-axis (k > 0) because y=x²+k. Hence, on the x-axis, the line can never be tangent to the parabola. DISCARD
Hence, cases 2 and 3 are the only plausible situations that can support st2, which leads to the conclusion that the point of intersection is in Q1.
Correct option: D

“A parabola in the coordinate geometry plane is represented by the equa GMAT data sufficiency" - is a topic of the GMAT data sufficiency section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT data sufficiency questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and two statements. By using mathematics to answer the question, the candidate must select the appropriate response among five choices which states which statement is sufficient to answer the problem. The data sufficiency section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

 

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