DS - XY plane with triangle

[karthikpandian19]

In the diagram above, the shaded region is bounded by, but does not include, the x-axis, the y-axis, and line l, which intersects the x- and y-axes at (b,0) and (0,a) respectively. If the equation of line l can be written as ry + 18x = 18b, does the point (1,2) lie inside the shaded region?

1. b/r = 1/6
2. ar = 27

[eagleeye]

In the diagram above, the shaded region is bounded by, but does not include, the x-axis, the y-axis, and line l, which intersects the x- and y-axes at (b,0) and (0,a) respectively. If the equation of line l can be written as ry + 18x = 18b, does the point (1,2) lie inside the shaded region?

1. b/r = 1/6
2. ar = 27

We are told that the equation of the line can be written as:
ry+18x = 18b
When x=0 => y = a = 18*(b/r), So a = 18*(b/r) or ar = 18b.
When y=0 => x = b
We can definitely determine whether (1,2) is inside the region if we know both x and y intercepts, (In other words, if we know both b and a). With that in mind, let's check the statements.

1. b/r = 1/6
We know that a = 18*b/r = 3.
So we know that a =3, we don't know what b is. INSUFFICIENT.

2. ar=27
We know that ar = 18b => 27 = 18b. b = 27/18 = 3/2 =1.5. We don't know what a is. INSUFFICIENT.

Together we know that a=3, b=1.5. So we can definitely find the relative position of (1,2) in the region. SUFFICIENT. Hence C.

Just as an aside, we didn't need to find whether (1,2) was inside the region, we just needed to be sure that "Data was sufficient". If we had to find it, This is how we would do it.
We have the equation of the line ( for x intercept b, and y intercept a) as:

x/b+y/a = 1

=> x/(3/2) + y/3 = 1
=> 2/3*x + y/3 = 1
If x = 1,
2/3 + y/3 = 1
y/3 = 1/3
y = 1
So (1,1) is on the line. Hence (1,2) would be above the line and therefore not in the region.

Let me know if this helps

[karthikpandian19]

See the colored portion below in your explanation:

In the diagram above, the shaded region is bounded by, but does not include, the x-axis, the y-axis, and line l, which intersects the x- and y-axes at (b,0) and (0,a) respectively. If the equation of line l can be written as ry + 18x = 18b, does the point (1,2) lie inside the shaded region?

1. b/r = 1/6
2. ar = 27

We are told that the equation of the line can be written as:
ry+18x = 18b
When x=0 => y = a = 18*(b/r), So a = 18*(b/r) or ar = 18b.
When y=0 => x = b
We can definitely determine whether (1,2) is inside the region if we know both x and y intercepts, (In other words, if we know both b and a). With that in mind, let's check the statements.

1. b/r = 6 I hope this a typo??
We know that a = 18*b/r = 3.
So we know that a =3, we don't know what b is. INSUFFICIENT.

2. ar=27
We know that ar = 18b => 27 = 18b. b = 27/18 = 3/2 =1.5. We don't know what a is. INSUFFICIENT.

Together we know that a=3, b=1.5. So we can definitely find the relative position of (1,2) in the region. SUFFICIENT. Hence C.

Just as an aside, we didn't need to find whether (1,2) was inside the region, we just needed to be sure that "Data was sufficient". If we had to find it, This is how we would do it.
We have the equation of the line ( for x intercept b, and y intercept a) as:

x/b+y/a = 1 ....How did you arrive at this equation????

=> x/(3/2) + y/3 = 1
=> 2/3*x + y/3 = 1
If x = 1,
2/3 + y/3 = 1
y/3 = 1/3
y = 1
So (1,1) is on the line. Hence (1,2) would be above the line and therefore not in the region.

Let me know if this helps

[eagleeye]

See the colored portion below in your explanation:

In the diagram above, the shaded region is bounded by, but does not include, the x-axis, the y-axis, and line l, which intersects the x- and y-axes at (b,0) and (0,a) respectively. If the equation of line l can be written as ry + 18x = 18b, does the point (1,2) lie inside the shaded region?

1. b/r = 1/6
2. ar = 27

We are told that the equation of the line can be written as:
ry+18x = 18b
When x=0 => y = a = 18*(b/r), So a = 18*(b/r) or ar = 18b.
When y=0 => x = b
We can definitely determine whether (1,2) is inside the region if we know both x and y intercepts, (In other words, if we know both b and a). With that in mind, let's check the statements.

1. b/r = 6 I hope this a typo??
We know that a = 18*b/r = 3.
So we know that a =3, we don't know what b is. INSUFFICIENT.

2. ar=27
We know that ar = 18b => 27 = 18b. b = 27/18 = 3/2 =1.5. We don't know what a is. INSUFFICIENT.

Together we know that a=3, b=1.5. So we can definitely find the relative position of (1,2) in the region. SUFFICIENT. Hence C.

Just as an aside, we didn't need to find whether (1,2) was inside the region, we just needed to be sure that "Data was sufficient". If we had to find it, This is how we would do it.
We have the equation of the line ( for x intercept b, and y intercept a) as:

x/b+y/a = 1 ....How did you arrive at this equation????

=> x/(3/2) + y/3 = 1
=> 2/3*x + y/3 = 1
If x = 1,
2/3 + y/3 = 1
y/3 = 1/3
y = 1
So (1,1) is on the line. Hence (1,2) would be above the line and therefore not in the region.

Let me know if this helps

1. b/r one was a typo. Clearly, it is 1/6.

2. Equation of a line can be written in many forms. One of them is called the intercept form.
The intercept form is as follows:

x/(x-intercept) + y/(y-intercept) = 1.
In this case, from the question image, we saw that x-intercept is b, and y-intercept is a. Hence we get:
x/b + y/a = 1.

[GMATGuruNY]

In the diagram above, the shaded region is bounded by, but does not include, the x-axis, the y-axis, and line l, which intersects the x- and y-axes at (b,0) and (0,a) respectively. If the equation of line l can be written as ry + 18x = 18b, does the point (1,2) lie inside the shaded region?

1. b/r = 1/6
2. ar = 27

ry + 18x = 18b
ry = -18x + 18b
y = (-18/r)x + 18b/r

Thus, the y-intercept = 18b/r.
Since the y-intercept is a:
a = 18b/r.

Statement 1: b/r = 1/6.
Since a = 18b/r, a = 18(1/6) = 3.
Thus, the y-intercept = (0,3).
INSUFFICIENT.

Statement 2: ar = 27.
Since a = 18b/r can be rephrased as ar = 18b, we get:
27 = 18b
b = 27/18 = 3/2.
Thus, the x-intercept = (3/2,0).
INSUFFICIENT.

Statements 1 and 2 combined:
Since we know both the y-intercept and the x-intercept of line l, we can determine whether (1,2) lies inside the shaded region.
SUFFICIENT.

The correct answer is C.

[karthikpandian19]

Oh gr8....you have dusted my mind out to remember one of the equation studied in school.

Thank you

See the colored portion below in your explanation:

In the diagram above, the shaded region is bounded by, but does not include, the x-axis, the y-axis, and line l, which intersects the x- and y-axes at (b,0) and (0,a) respectively. If the equation of line l can be written as ry + 18x = 18b, does the point (1,2) lie inside the shaded region?

1. b/r = 1/6
2. ar = 27

We are told that the equation of the line can be written as:
ry+18x = 18b
When x=0 => y = a = 18*(b/r), So a = 18*(b/r) or ar = 18b.
When y=0 => x = b
We can definitely determine whether (1,2) is inside the region if we know both x and y intercepts, (In other words, if we know both b and a). With that in mind, let's check the statements.

1. b/r = 6 I hope this a typo??
We know that a = 18*b/r = 3.
So we know that a =3, we don't know what b is. INSUFFICIENT.

2. ar=27
We know that ar = 18b => 27 = 18b. b = 27/18 = 3/2 =1.5. We don't know what a is. INSUFFICIENT.

Together we know that a=3, b=1.5. So we can definitely find the relative position of (1,2) in the region. SUFFICIENT. Hence C.

Just as an aside, we didn't need to find whether (1,2) was inside the region, we just needed to be sure that "Data was sufficient". If we had to find it, This is how we would do it.
We have the equation of the line ( for x intercept b, and y intercept a) as:

x/b+y/a = 1 ....How did you arrive at this equation????

=> x/(3/2) + y/3 = 1
=> 2/3*x + y/3 = 1
If x = 1,
2/3 + y/3 = 1
y/3 = 1/3
y = 1
So (1,1) is on the line. Hence (1,2) would be above the line and therefore not in the region.

Let me know if this helps

1. b/r one was a typo. Clearly, it is 1/6.

2. Equation of a line can be written in many forms. One of them is called the intercept form.
The intercept form is as follows:

x/(x-intercept) + y/(y-intercept) = 1.
In this case, from the question image, we saw that x-intercept is b, and y-intercept is a. Hence we get:
x/b + y/a = 1.