Lines a and b have different y-intercepts. When line a is

Question

Lines a and b have different y-intercepts. When line a is reflected around the y-axis, is its reflection parallel to line b?

(1) Line a is perpendicular to line b.

(2) The slope of line b > 0.

KarishmaB · Accepted Answer

Upon request, I am providing the solution of the question above:
I will provide a graphical approach since that is what I favor always but later will give an algebraic approach too...

The two diagrams below illustrate the case where we take both statements together. In one case, reflection of a is not parallel to b and in the other reflection of a is parallel to b. Hence even with both statements we cannot say whether reflection of a is parallel to b. Answer (E).

Attachment:

Ques2.jpg [ 23.2 KiB | Viewed 8364 times ]

If a and b make 45 degrees angle with the y axis (as shown, technically I will not say that they are both making 45 degrees angle with y axis but let's not worry about it here), when a is reflected along y axis, its angle with y axis is still 45. In this case a and b are parallel.

Algebraic approach:
A line is defined by 2 things - its slope and y intercept. When we reflect a line along the y axis, its slope flips sign but y intercept remains unchanged.

Now,
a -> y = mx + c
b -> y = nx + d
Reflected a -> y = -mx + c
Ques: Is -m = n? (Parallel lines have the same slope.)

Stmnt 1: mn = -1
m = -1/n.
If m = 1 and n = -1, -m is equal to n
If m = -1 and n = 1, -m is equal to n
If m = -1/2 and n = 2, -m is not equal to n
Not sufficient.

Stmnt 2: n>0
If m = -1 and n = 1, -m is equal to n
If m = -1/2 and n = 2, -m is not equal to n
Not sufficient.

Taking both together,
If m = -1 and n = 1, -m is equal to n
If m = -1/2 and n = 2, -m is not equal to n
Not sufficient. Answer (E).

gmat1220 · Answer

S1 insufficient
Y axis has no reflection about y axis. Assume line a= y axis. The answer is NO

If line a has slope -1. Slop of b is 1. Reflection of line a about y axis is parallel to line b. The answer is YES

S2 insufficient. 
Slope of line a is unknown

1) + 2) sufficient
Since b>0 the line b cannot be x axis. Line a cannot be y axis - a is perpendicular to b. That means when a has reflection about y axis it is parallel to b. The answer to the question is YES

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gmat1220 · Answer

But my initial guess was A alone sine we know that line a and line b have known and "different y intercepts". This precludes a from being the y axis . So I have two answers a or c. My bet 50/50 on both :-)

tough one!

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tough one! Posted from my mobile device

bhandariavi · Answer

But according to the 'bagrettin' explanation to my previous question:
a --- y = mx + b
b --- y = (-1/m)x + b
Reflection of line a --- x = ym + c ( shouldn't we flip x and y values to find reflection?)
 Y = (1/m) x - ( c/ m )
So now slope of line b is (-1/m) & slope of reflected line is (1/m)
It looks like they will never be parallel because if one is + ve another is - ve.
Help me ? I don't know whether we should proceed this way or not.

gmat1220 · Answer

Hello bhandariavi
x coordinate flips sign due to reflection about y axis. Draw two perpendicular lines in any quadrant and you will see that reflection of line a (whose slope -1) Reflected line will have slope = 1 is parallel to line b (slope 1) - ( knowing that line a is perpendicular to b) Visualize the problem that is easier than algebra :-)

or you can request bagrettin for algebraic solution. Cheers

Posted from my mobile device

or you can request bagrettin for algebraic solution. Cheers Posted from my mobile device

Onell · Answer

bhandariavi,
For reflection on y- axis you don't have to flip x and y values .Just negate x-coordinate
The reflection of the point (x, y) across the y-axis is the point (-x, y).
The reflection of the point (x, y) across the line y = x is the point (y, x).
The reflection of the point (x, y) across the line y = -x is the point (-y, -x). 
The reflection of the point (x, y) across the x-axis is the point (x, -y).

Onell · Answer

a --- y = mx + b
b --- y = (m1)x + b
Reflection of line a --- y = -mx + c (Reflection over Y-axis)
we got to find if (-m)= (m1)

Statement A : m *m1=-1..
m=-1/m1
M1=-1 AND M=1, 1=1 no
M1=1 AND M=1, -1=1 yes
So InSufficient...

Statement B b>0... Insufficient...

So It should be E....

gmat1220 · Answer

Hello onell
Due to reflection about y axis Not only the x flips sign but the slope of the line "also" flips sign. I believe your equation did not account for slope change. Pls correct me if this not true.

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Onell · Answer

If it flips the sign of x coordinate and changes the sign of a slope : You will get a original line
Consider a line passes through (x ,y)
 y = mx + b

Upon reflection it passes through (-x ,y) and changes slope to -m (As you have written)
 y = -m (-x) + c
 y=mx +c (Equation of a original line)
Am I missing sth ?

gmat1220 · Answer

Hello onell
Strange! Let me try will coordinates. Let's say line a (slope 1) passes through (2,3) and upon reflection passes through (-2,3) slope=-1
Equation of line a (slope 1) is
 (y-3)/(x-2) =1
y-3 =x-2
y=x+1 -----------(1)

The equation of the reflected line (slope -1) is
 (y-3)/ =-1
y-3=-1(x+2)
y-3 = -x - 2
y = -x + 1 ----------(2)

Two different equations. The slope of the line b which is perpendicular to a is -1. The slope of the reflected line is also -1. Hence E cannot be the answer.

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Onell · Answer

Strange Indeed. However if you substitute (2,3) and slope 1 and(-2,3) and slope -1 for reflected line in equation y=mx+c . You get the same equation for both the line..
 Bunuel, Please help....

sudhir18n · Answer

line a = y =mx+b
line b = y = nx+c

now in case a line reflects, its slope changes sign and Y intercept remains same
so relected a = y= -mx+b

now the question asks is -m =n ( parallel lines will have same slope)

st1. -m*n = -1
 m = 1 and n = 1 ... satisfies our condition
 m = 1/3 and n = 3 ; -m*n = -1 but -m is not equal to n ... hence in sufficient

st2: n>0
 same as above... insufficient

1&2 
 m = 1 n =1 ok 
m = 1/3 and n = 3 not ok

hence E

gnan · Answer

let the lines be 
a ---> y = mx + c
b ---> y = nx + d

reflection of line 'a' around y-axis would be y = - mx + c

from the first statement,
mn = - 1

but we cannot decide whether they are parellel as '- m' can be or cannot be equal to 'n'.

from the second statement,

n>0

this signifies nothing regarding the relationship between slopes of the two lines, hence not sufficient

Even after combining both the statements, the data is insufficient

For example, if m = -1; n = 1
it satisfies both the statements and the lines are parellel

and for m = -0.5; n = 2
it satisfies both the equations and the lines are not parellel.

Ans. E

KarishmaB · Answer

The answer here is (E). I have provided the graphical and algebraic approaches here: lines-a-and-b-have-different-y-intercepts-when-line-a-is-111182.html#p897728

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