BTGModeratorVI wrote: ↑Wed Feb 03, 2021 10:29 am
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.
Answer:
D
Source: Veritas Prep
Given: 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.
Target question: Is the point of intersection in Quadrant I?
First off, here's what the graph of y = x² looks like:
Since we're adding
k (which is positive) all of the y-coordinates of the graph of y = x² will be increase by
k units.
So the graph of y = x² +
k will look something like this:
Important: Since the graph of y = x² +
k lies in quadrants I and II only,
the point of intersection (of the line and parabola) must be in EITHER quadrant I OR quadrant II)
Statement 1: The slope of line L is positive
This statement is sufficient. Here's why:
If a line of positive slope intersects the parabola in quadrant II, then that line will also intersect the parabola at a point in quadrant I.
For example:
Since we're told line L intersects this parabola at exactly one point, we can be certain that
line L can't intersect the parabola in quadrant II
This means
line L must intersect the parabola in quadrant I only
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
If you're not convinced, here's another way to look at it.
If line L intersects this parabola at
exactly one point, then line L must be
tangent to the parabola.
So for example, line L might look like this:
Or like this:
As you can see, if line L has a positive slope
and is tangent to the parabola,
the point of intersection must lie in quadrant I
Statement 2: x is greater than 0
I'm not crazy about this wording.
For clarity, I would write statement 2 as follows:
The x-coordinate of the point of intersection is positive
This means line L must be tangent to the parabola at one of the
red dots shown below
Since all of the possible points of intersection are in quadrant I, we can be certain that
line L must intersect the parabola in quadrant I
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: D
Cheers,
Brent
