Data Sufficiency

Is a<2?

1. In the xy-plane, the point (a, 1) lies inside the circle whose equation is x^2 + y^2 = 3.

2. In the xy-plane, the point (a, 4) lies on the line whose equation is 2y+4x = 10.

[spoiler]OA: D

Source: GMATPrep Exam Pack 1

[/spoiler]

nataras wrote:Is a<2?

1. In the xy-plane, the point (a, 1) lies inside the circle whose equation is x^2 + y^2 = 3.

2. In the xy-plane, the point (a, 4) lies on the line whose equation is 2y+4x = 10.

The answer to the question stem can be NO only if a is positive.
Thus, when we evaluate the statements, we need consider only POSITIVE values for a.

Statement 1: In the xy-plane, the point (a, 1) lies inside the circle whose equation is x^2 + y^2 = 3.
Every point (x, y) such that x² + y² = 3 lies ON the circle.
Every point (x, y) such that x² + y² > 3 lies OUTSIDE the circle.
Every point (x, y) such that x² + y² < 3 lies INSIDE the circle.

Since (a, 1) must lie INSIDE the circle, (a, 1) must satisfy x² + y² < 3:
a² + 1² < 3
a² < 2
a < √2.
Thus, a < 2.
SUFFICIENT.

Statement 2: In the xy-plane, the point (a, 4) lies on the line whose equation is 2y+4x = 10.
Substituting x=a and y=4 into 2y+4x = 10, we get:
2(4) + 4a = 10
4a = 2
a = 2/4 = 1/2.
Thus, a < 2.
SUFFICIENT.

The correct answer is D.

Is a<2?

1. In the xy-plane, the point (a, 1) lies inside the circle whose equation is x^2 + y^2 = 3.

2. In the xy-plane, the point (a, 4) lies on the line whose equation is 2y+4x = 10.

Using Statement 1)
If (a,1) is inside the circle then the distance of (a,1) from the centre of circle should be less than radius (√3)
i.e. (a^2) + (1^2) < 3
i.e. a^2 < (3-1)
i.e. a^2 < 2
[spoiler]i.e. -√2 < a < √2 SUFFICIENT[/spoiler]

Using Statement 2)
If the point (a, 4) lies on the line whose equation is 2y+4x = 10 then the point will satisfy the above equation
i.e. 2 x(4) + 4 x (a) = 10
[spoiler]i.e. a= 2/4 = 0.5 Which is smaller than 2. Therefore, SUFFICIENT[/spoiler]

[spoiler]Correct Answer Option - D[/spoiler]

GMATGuruNY Quoted:
a² < 2
a < √2

Hi GMATGuruNY,

Shouldn't it be corrected to
a² < 2
-√2 < a < √2

Because nowhere is it mentioned that a is positive.

Although it doesn't change the answer of this particular question but somewhere else it may.

GMATinsight wrote:

GMATGuruNY Quoted:
a² < 2
a < √2

Hi GMATGuruNY,

Shouldn't it be corrected to
a² < 2
-√2 < a < √2

Because nowhere is it mentioned that a is positive.

Although it doesn't change the answer of this particular question but somewhere else it may.

The answer to the question stem can be NO only if a is positive.
Thus, when we evaluate the statements, we need consider only positive values for a.
I've amended my post above to make the reasoning clear.

Just to add two time saving tips for this question:

Given S1, we know that any point (x,y) inside the circle will satisfy the equation x² + y² < 3. Since the maximum value of x comes when y = 0, the greatest x we can have = √3. So we KNOW a < √3 < 2, and we don't need to do any substitution.

Given S2, we have two equations (2y + 4x = 10 and y = 4) and two variables, so we KNOW we can solve without bothering to.

This strikes me as a question differentiating between those who are comfortable with the coordinate plane (and will need virtually no time to solve it) and those who aren't (who will have a horrible time solving it, or even understanding the implications of the two statements). The testwriters seem to be asking tricky and novel (by their standards) coordinate geometry questions lately, so anyone reading this who feels weak in this subject is advised to brush up.