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# At what angle do line y=Kx + B and line y=Bx+K intersect? 1.

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CEO
Joined: 21 Jan 2007
Posts: 2764
Location: New York City
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Kudos [?]: 350 [0], given: 4

At what angle do line y=Kx + B and line y=Bx+K intersect? 1. [#permalink]  19 Nov 2007, 07:32
At what angle do line y=Kx + B and line y=Bx+K intersect?

1. B+K=1
2. B*K=0

Please walk me thru this problem.
CEO
Joined: 17 Nov 2007
Posts: 3578
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 407

Kudos [?]: 2149 [0], given: 359

Expert's post
C.

angle(kx+b,bx+k)=angle(kx,bx) y=kx+b (k: slope, b: shift)
k,b - characterize slopes of lines.

1. b+k=1
for example, (b,k)=(0,1) ==> angle=45
(b,k)=(1000001,-1000000) ==> angle~0
insuff.

2. b*k=0
for example, (b,k)=(0,1) ==> angle=45
(b,k)=(0,-1000000) ==> angle~90
insuff.

3. b+k=1, b*k=0 ==> (b,k) e {(0,1);(1,0)}
for both cases angle=45
suff.
Manager
Joined: 25 Jul 2007
Posts: 108
Followers: 1

Kudos [?]: 14 [0], given: 0

Answer is C. Both statements together are sufficient.

Here is a detailed explanation.

Consider statement 1.

B + K =1.

Substituting this equation in the 2 given equations doesn't tell us much.

Statement 1 is insufficient.

Consider statement 2.

B * K =0.
i.e. either b=0 or k=0 or b=0,k=0.

This does not give us specific information about the position of the lines either.

Consider the two statements together.

B + K =1

&

B * K =0.

That means either B = 0 and K = 1

OR

B = 1 and K = 0.

Considering either of the possibilities in the given line equations leaves us with the same two equations.

i.e y = 1 and y = x

y = 1 is a line parallel to the axis at a height of 1.

y=x will have points such as (0,0), (1,1), (2,2),....

Therefore, these two lines can only intersect at one point (1,1).

We now have an isosceles right angled triangle.

i.e. the two lines intersect at an angle of 45 degrees.

Hope this helps.
CEO
Joined: 21 Jan 2007
Posts: 2764
Location: New York City
Followers: 9

Kudos [?]: 350 [0], given: 4

good job guys.

OA is C.

slope of 1 is 45 deg
slope of 0 is flat, parallel to x axis
intersect at 45 deg.
Director
Joined: 11 Jun 2007
Posts: 649
Followers: 3

Kudos [?]: 84 [0], given: 0

Another way (more useful for arbitrary angles):
easy formula for any general straight lines:

The angle between two lines y = m1x + c1 and y = m2x + c2 is tan inverse of the modulus of : [(m1 - m2)/(1 + m1*m2)]

tan X = ( B - K)/(1 + B*K)

case 1: tan X = ( B - K)/(1 + B*K) ( not sufficient)

case 2: tan X = (B - K) (not sufficient)

combine case 1 and 2:

tan X = (B - K) where K = 0 or B = 0. Therefore, B = 1 or K = 1 respectively.

Hence tan X = 1;(taking the positive angle) X = 45 degrees.
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