On the x-y coordinate grid, does the line passing through point P (not

Question

On the x-y coordinate grid, does the line passing through point P (not shown) have a slope greater than 2?

(1) Point P has an x-value of 1 and is 2 units away from the origin
(2) The line does not pass through Quadrant IV

nick1816 · Answer

Statement 1- P(x,y) = (1,y)

x^2+y^2 = 4

y= ±\sqrt{3}

P can be  (1,\sqrt{3})  or  (1,-\sqrt{3})

We have no idea about slope though.

Insufficient

Statement 2 -

Slope of y axis is infinite.
Slope of x-axis is 0.

Clearly insufficient

Combining both statements

P(x,y)=  (1,\sqrt{3})

Disregard other case as  (1,-\sqrt{3}) ) lies in Quadrant IV.

Now we can clearly see in the diagram that maximum possible slope of a line that contains P must pass through origin.
Also. minimum possible slope of a line that contains P must be parallel to x-axis)

Maxm possible slope=  \frac{\sqrt{3}-0}{1-0}  =  \sqrt{3})  <2

Sufficient

CRACKGMATNUT · Answer

Hi,

Even if we take scenario 2  (1,-\sqrt{3}) , then also slope should be negative, which will be less than 2. 
Shouldn't St 1 be sufficient?

Fdambro294 · Answer

Line passes through point P

We want to know if the slope passing through this Point P is > 2 ?

S1: if the X coordinate is 1 and Point P is 2 units away from the origin, by setting up a right triangle in the coordinate plane we can determine the unique coordinates of Point P.

P (1 , y)

(1)^2 + (y)^2 = (2)^2

y- coordinate = +/- sqrt(3)

Which means point P is located either in Quadrant I or Quadrant IV

However we do not know from which angle or direction the Line crosses through either point.

The Line could be downward sloping with a negative slope (NO)

Or it could be upward sloping with a much steeper positive slope (YES)

S1 not sufficient

(S2) the line does not pass through Quadrant IV

Rule: a (-)negative sloping line MUST ALWAYS, at the very least, pass through Quadrants II and IV

It also may or may not pass through one of either Quadrant I or Quadrant III

Since the line can not pass through Quadrant IV, we know that the slope of the Line must be (+)positive. But, we do not know if the slope is greater than > 2

S2 not sufficient

(1) + (2)

From statement 1, we know that the Line must pass through point P which is located at either:

(1 , sqrt(3) )

Or

(1 , (-)sqrt(3) ) ———-> however, since this point would be located in Quadrant IV, this value is impossible for Point P

Thus, from both statements, we know that the Line has a (+)positive slope and must intersect Point (1 , sqrt(3)) located in Quadrant I

Is the slope > 2?

The key condition is that the Line can not pass through Quadrant IV.

Thus, the set of possible lines that meet this condition AND pass through Point P range from:

Y = sqrt(3) ————> a Horizontal Line with slope 0

And

The steepest line we can create that passes through a Point P located in Quadrant I and does NOT pass through Quadrant IV ———> will have its Y intercept located at the Origin (0 , 0)

If we make the Y Intercept any negative value (which would result in a steeper line) we would then violate the condition that the line can not pass through Quadrant IV

Therefore, the MAX slope we can have for the Line is where it passes through:

(0 , 0) and (1 , sqrt(3))

The slope for such a line would be = sqrt(3) / 1 ————> which is STILL LESS THAN < 2

Definite NO answer and Together (1) + (2) are sufficient

C

Posted from my mobile device

Fdambro294 · Answer

Wherever the point P is located, we do not know the angle from which the Line is coming in to pass through Point P.

For statement 1, if the point P is located at the point you listed in Quadrant IV, the Y-Intercept could be an extremely high magnitude (-)negative Y Coordinate ———> perhaps Y-Intercept = -100,000

in that case, you can imagine a Line with an extremely steep (+)positive slope that would be > 2

On the other hand, if the Line has a positive Y Intercept, then the Line will be downward sloping. The line will have a (-)negative slope.

The Line could even pass through the point horizontally and have a slope of 0

That is why statements 1 is NOT sufficient.

Hopefully that helped somewhat?

Posted from my mobile device

bumpbot · Answer

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On the x-y coordinate grid, does the line passing through point P (not

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On the x-y coordinate grid, does the line passing through point P (not

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