If lines y=mx+b and x=y+bm intersect at a degrees angle

To find the angle between two lines, we need to know the slope of both lines. But as shown in the figure, this angle "a" can be either "x" or "y". But since we are given that a<90, we can find out which angle is required because x + y = 180. The slope of the second line is obviously 1. So the question is basically asking for the value of m.

1) Sufficient

2) We get y = bx + b. b is still unknown. Insufficient.

Answer is hence A.

Kudos Please... If my post helped.

We do not care about the actual measure of the angle.

When we have to determine the angle at which two lines intercept, the ONLY thing we have to know is the slope of each line .
With statement 1 we get (note that I consider only the slopes of the two equations):
line1: \(y=2x\)
line2: \(y=x\)

Once we have those, we are able to determine the length of the angles at the point of intersection. Those angles are fixed and do not change, hence we can determine their length, but in this DS we do not care about the actual number.
With statement 1, can you answer the question? YES, that's enough.

To determine the angle we would have to use formulas that are beyond the scope of the GMAT ("tan" for example) https://planetmath.org/anglebetweentwolines
but the point here, as I said above, is not to find the measure.

We can find the angle of intersection b/w any 2 lines if we knw the values of their individual slopes.

For y = x + bm, slope is 1; for y = mx + b, slope is "m"
(1) Tan (a) = (m - 1)/(1 + 1*m); tan(a) < 90 if its value is +ve;since m-1>0, no need to add/subtract from 180.
Since statement 2 gives m=2, it is sufficient.
(2) Test y=x+1, y=2x+2 and y=3x+3 with "y=x - 1, y=x-4, and y=x-9"
y=x+1 and y=x gives 0 degrees
y=2x+2 and y=x-4 gives a diffrent value...INSUFF

Folks,
I understand A is sufficient to find the angle.. but how do you find the angle?
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m= tan x.
So x= tan inverse(m).

but m is slope of which line out of the that intersect?
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m is the slope of the line
y=mx+b

If you draw an equation for this line, you will find m to be the slope of the line and b to be the intersect on y axis(when x=0). This is called the slope-intercept form of line equation and you memorizing it will help you deal with such questions. The form of such lines is
y=(slope)x+y-intersect

The other line x=y+bm can be written in a similar fashion

y=x-bm.

Going by the above stated formula, since the coefficient of x=1, slope =1. The y-intersect of the line is bm.

Hope that clarifies your doubt.

Angle between 2 lines is m1-m2/1+m1m2 ..

We already know slope of line X = Y + bm.

Option A tells slope of line A - so suffcient but option B tells nothing so not sufficient

How do we apply this formula?
In this question m1= 2 and m2=1, does that mean angle is \(\frac{2-1}{1+2.1} = \frac{1}{3}\)
is there a \(tan^{-1}\)before this, so angle between two lines having slopes m1 and m2 \(= tan^{-1}\frac{m1-m2}{1+m1m2}\)

In short how do we actually determine the angle between 2 lines given their slopes?
Thanks
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That definitely helps ! Other solutions involving tan kept me wondering if indeed it was beyond scope or not,+1
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Correct answer is A

Since Slope of y=mx+b is m let it be m1
and slope of y=x-mb is 1 , let it m2

Now
tan(a) = m1-m2/(1+m1*m2)

ie tan(a) = m-1/(1+m*1)= m-1/m+1


Hence knowing the value m will give us answer

if lines y = mx + b and y = x + bm intersect at A degrees (A<90) what is the value of angle A

1) m = 2
2) m = b
This is a DS question

The Question has been discussed before: if-lines-y-mx-b-and-x-y-bm-intersect-at-a-degrees-angle-142552.html#p1144329

Hello akshaybansal1991,

Welcome to GC!!

You may want refer to below posts for easier navigation on the site and get more out of Gmatclub

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The Question has been discussed before: if-lines-y-mx-b-and-x-y-bm-intersect-at-a-degrees-angle-142552.html#p1144329

Coming back to the question
if lines y = mx + b and y = x + bm intersect at A degrees (A<90) what is the value of angle A

1) m = 2
2) m = b

We know the slope of lines as 1 and m and thus we need to find only m and we can find the angle between 2 lines.

St1 gives you that data and is sufficient
St2 gives you nothing about value of m and hence not sufficient.

Similar question practice
lines-intersecting-angles-m08q15-66826.html#p1235093
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Agree that Tan and other trignometry concepts are out of the remit of GMAT but is the formula for angle between two lines m1-m2/1+m1m2 in scope ? In other words can this question come in PS section ?

Given : 2 lines y = mx+b and y = x-bm. THey interesect at an angle a. Also a < 90
DS : Value of angle a.

a<90, so we know that we need to find only acute angle between the two lines.

Statement 1 : m =2, y = 2x +b, y = x - 2b

So we know the slopes of the two lines and hence the angle between them can be easily found out by the slope fomula
angle a = \(tan^-1 \frac{{m1-m2}}{{1+m1*m2}}\)
SUFFICIENT

Statement 2: m = b, y = bx +b , y = x- b^2

Here the slopes are b and 1 respectively for the two given lines. So we can't find the slope between the two lines.
NOT SUFFICIENT

Answer A

Hi All,

We're told that the lines Y = (M)(X) + B and X = Y+ (B)(M) intersect at A degrees angle (where A <90 ). We're asked for the value of angle A. To start, this is a complicated-looking graphing question, but ass a category, you're likely to see just 1-2 graphing questions on Test Day and they're normally not this complex.

Since the question asks for the measure of angle A (the smaller angle that forms when the two lines intersect), I would make sure that the lines were in slope-intercept format:

Y = (M)(X) + B
Y = X - (M)(B)

Since we're interested in the angle that forms, we need to focus on the SLOPE of the two lines (the Y-intercept would only determine where each line crosses the Y-axis).

1) M = 2

Based on the information in Fact 1, the lines would be....

Y = 2X + B
Y = X - 2B

No matter what values you choose for B (and then for X and Y), the two lines would have consistent slopes (so they'd always be at the same respective angles, just on different parts of the graph). Try drawing some example pictures and you'll see). This means that the lines will intersect in the exact same way every time and that angle A will always be the same.
Fact 1 is SUFFICIENT.

2) M = B

Since M = B, they could both be 2, 3, 4, 5, etc.

This changes the slope of the first line (depending on the slope), while the slope of the second line stays the same. So, the angle between the two lines would change. If you draw two example pictures, then you see that angle A changes.
Fact 2 is INSUFFICIENT.

Final Answer:

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