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## GMAT - Data Sufficiency - Coordinate geometry

### In the xy-plane, region R consists of all the points (x,y) such that 2x+3y?6. Is the point (r,s) in region R? (more)

In the xy-plane, region R consists of all the points (x,y) such that 2x 3y≤6. Is the point (r,s) in region R?
(1) 3r 2s=6
(2) r3 and s2

the point is (r, s)

given region R: 2x + 3y <= 6

we represent region R as shaded region in the figure below the line 2x + 3y = 6

Statement 1

3r+2s=6

since r and s are x and y coordinates respectively, (r, s) must lie on the line

3x + 2y = 6

Both lines interesect, and have different slopes, hence some part of the line will lie above the region and some part of the line will lie below the region.

So we cannot say with certainity that the point will satisfy the condition.

Insufficient

Statement 2

r3 and s2

plotting x =3 and y = 2

All the points below these curve will represent r <= 3 and s <= 2

We see that points like (3, 2) lie outside the shaded region.

Insufficient

### In the xy-coordinate plane, is point R equidistant from (-3, -3) and (1, -3) (more)

In the xy-coordinate plane, is point R equidistant from (-3, -3) and (1, -3)

Statement 1:

X coordinate of R = -1

Statement 2:

R lies on the line y =  -3

Statement 1:

let's say R = (-1, y)

distance of R from (-3, -3)

$\sqrt { { (-3-(-1)) }^{ 2 }+{ (-3-y) }^{ 2 } } =\quad \sqrt { 16\quad +\quad 9\quad +\quad 6y\quad +\quad { y }^{ 2 } } =\quad \sqrt { { y }^{ 2 }+6y+25 }$

distance of R from (1, -3)

$\sqrt { { (1-(-1)) }^{ 2 }+{ (-3-y) }^{ 2 } } =\quad \sqrt { 4\quad +\quad 9\quad +\quad 6y\quad +\quad { y }^{ 2 } } =\quad \sqrt { { y }^{ 2 }+6y+13 }$

The distances can never be equal because

$\quad \sqrt { { y }^{ 2 }+6y+25 }$ can never be equal to $\quad \sqrt { { y }^{ 2 }+6y+13 }$

thus statement 1 alone is sufficient.