Does the line L intersect the graph of the curve y = x^2 – 6?

Question

(1) The line L passes through the origin and makes an angle of 45o with the positive x-axis.

(2) The line L passes through the point (-2, -2) and (2, 2).

GMATinsight · Accepted Answer

The best ways to draw the graphs is substituting values of x and y

e.g. in Equation y = x^2 - 6

for x = 0, y = -6

for y = 0, x^2 = 6 i.e. x =  +  \sqrt{6} i.e. x =  + 2.5 (approx.)

Now join the points obtained with free hand and you get the parabola.

Press +1Kudo if you like my post

GMATinsight · Answer

The best Method to solve this question will be plotting it graphically

The Graph of y = x^2 - 6 is as follows

The Statement 1 says a line passing through origin at 45 degrees i.e. equation of line y=x which crosses the primary graph at two points as shown in the second picture
Therefore SUFFICIENT

The Statement 2 says a line passing through (-2, -2) and (2, 2) which has equation of line y=x which crosses the primary graph at two points as shown in the second picture
Therefore SUFFICIENT

Answer: Option  D

TARGET730 · Answer

GMATInsight, Thanks for your quick response. I am not a quant whiz

Can you please explain how to draw the graph for the equation y = x^2 - 6 ?

TARGET730 · Answer

Great. Thanks much

Temurkhon · Answer

All we need to know is only two points of L or one point and slope. Wa can ignore any information about parabola

St1. Point (0;0) and slope=1. SUFF

St.2. Two points given, so we can count slope. SUFF

D

KarishmaB · Answer

Note here that all you need to worry about is getting a unique line. If you have the equation of a line and that of a curve, you will be able to say whether the line intersects the curve or not. The point is that it is not important to figure out whether it actually does intersect because in either case, the information will be sufficient - irrespective of whether the answer is yes or no. So actually drawing the curve and the line is not required.

Both statements individually give you a distinct line. So you will be able to say whether the line intersects the curve or not.

Answer (D)

This brings up an interesting question - given the equation of the curve y = x^2 – 6, what kind of lines will not intersect it? Think about it.

KarishmaB · Answer

It will be great if you learn how to quickly draw and manipulate graphs of lines and some curves such as circle and parabola. Check a discussion on lines and points here: https://www.youtube.com/watch?v=3kX5UtvHGFg You can follow the same approach with curves if you know some basic shapes such as y = x^2 is an upward opening parabola centered at (0, 0). Check how quadratic equations represent parabolas here: https://www.youtube.com/watch?v=QOSVZ7JLuH0

TARGET730 · Answer

Thank you Karishma

anmolgmat14 · Answer

I solved the question by assuming that both the statements mentioned tells us about that line passes through origin.
And then solved the equation of parabola and checking whether discriminant equals to zero or not.

Since it is not equals to zero it will not intersect.

Please help and correct me, as i am new to this topic and getting confused while solving Coordinate Geometry Questions. 
Also, if someone can share some study material on coordinate geometry

KarishmaB · Answer

I am not sure what you mean by solving the equation of parabola. It is already given to you. Do you mean you found the equation of the line and then solved with the equation of parabola to see if they intersect? As I mentioned earlier, solving to find the points of intersection is useless. As long as we get unique equations for line and parabola, we CAN find and answer. Hence it WILL be sufficient.

funkyakki · Answer

I can't figure out how to find out the equation for the line from statement 1. Please help

GMATinsight · Answer

Hi  funkyakki

The 'Statement 1' has only one information that line is making 45º with positive x-axis i.e. Slope of the line = 1 because for a line making 45º,  y_2-y_1 = x_2-x_1 )

Since the line is passing through origin therefore y-intercept of line = 0

The equation of line is given by the equation   y = mx+c

where m = Slope = 1
c = y-Intercept = 0

i.e. Equation of line becomes   y = x

Check teh graph as illustrated by me in other explanation on this thread

https://gmatclub.com/forum/download/file.php?id=25568

GMATBusters · Answer

y = x^2 – 6

Equation of curve is given the graph can be plotted uniquely.

(1) The line L passes through the origin and makes an angle of 45 Deg with the positive x-axis.
 Slope and one point (0,0) given. so the line can be drawn uniquely.

Both Curve and line can be plotted to find the point(s) of intersection.
SUFFICIENT,

(2) The line L passes through the point (-2, -2) and (2, 2).
 Two points on the line are given, so the line can be drawn uniquely.

Both Curve and line can be plotted to find the point(s) of intersection.
SUFFICIENT.

Answer D

(PS: there is no need to find the equation of line or try to find the point(s) of intersection)

bumpbot · Answer

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Does the line L intersect the graph of the curve y = x^2 – 6?

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Does the line L intersect the graph of the curve y = x^2 – 6?

Prep Toolkit

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