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A line with the equation y = px + q is reflected over the

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A line with the equation y = px + q is reflected over the [#permalink] New post 20 Mar 2011, 17:14
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A line with the equation y = px + q is reflected over the line y = x. Is the reflection of this line parallel to the line y = mx + n?

(1) m = p + 2

(2) m = 3p

I didn't get this question though got the answer right :)
What exactly does reflection mean?
[Reveal] Spoiler: OA

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Re: line equation????????? [#permalink] New post 20 Mar 2011, 23:38
S1) insufficient. Slopes are unknown
S2) insufficient
Consider line m to be inclined 60 deg to x axis and line p to be inclined 30 deg to x axis. The reflection of line p about y=x is parallel to line m. The answer to the question is yes.
Consider line m and line p to be parallel to x axis. The reflection of line p about x=y is not prallel to the line m. The answer to the question is no

1) + 2) sufficient
The slopes of the lines are known m=3 and p=1. we can know the angle each line makes with the x axis. And know for sure whether the reflection of line p is symmetrical about the x=y. This can be definitely answered. Sufficient. Hence C

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Last edited by gmat1220 on 21 Mar 2011, 00:12, edited 1 time in total.
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Re: line equation????????? [#permalink] New post 21 Mar 2011, 00:10
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Any monotonous function f(x) which is reflected over the line y=x becomes the function g(x) which is the inverse to f(x). This means f(g(y))=y for all y from the domain and g(f(x))=x for all x from the domain. In case of linear function it is very easy to find the inserve just by changing y to x and x to y.
In our case the inverse x=py+q
y=x/p-q/p

You can read more about inverse functions here
http://en.wikipedia.org/wiki/Inverse_function

If this reflection is paralle to the line y=mx+n, then the slope of two lines are equal, therefore m=1/p. So, finally the question asks whether m=1/p.

(1) says m=p+2, which means that m=1/p only for some particular values of p, so we could not conclude whether the lines are parallel
(2) says m=3p, which means that m=1/p only for some particular values of p, so we could not conclude whether the lines are parallel

(1) and (2) gives us the system
m=p+2
m=3p

from which p+2=3p
p=1 and m=2+1=3

As we could see, m is not equal to 1/p, so these lines are not parallel. Only both statements together are sufficient for the answer.

The answer is C.
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Re: line equation????????? [#permalink] New post 21 Mar 2011, 00:15
@bagrettin pls verify my reasoning. Thanks for confirming the answer

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Re: line equation????????? [#permalink] New post 21 Mar 2011, 01:21
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Hello, gmat1220 :)

Your reasoning is right!

In S1. Just to be more precise, let say that correspondence between slopes p and p+2 are different for the different values of p, therefore the answer on the question will be different too (for exact equation, linking inverse function see my post)

In S2 your example with 60 and 30 degrees corresponds to m=3p. The line which is inclined 60 degrees to x axis has the slope of 3^0.5 (tg 60), while the line with 30 degrees to x axis has the slope 1/3^0.5 (tg 30), so m=3p. Again since y=x is inclined at 45 degrees to Ox, this lines are symmetrical to the y=x.

If both lines have the slope of 0, then m=3p, while the reflection will be vertical line with the slope equal to infinity.
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Re: line equation????????? [#permalink] New post 21 Mar 2011, 07:20
Hello bagrettin
Your explanation rocks!!!!! Thanks so much.

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Re: line equation????????? [#permalink] New post 21 Mar 2011, 13:03
Hello, gmat1220!

Happy to help you) You are always welcome! ;)
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Re: line equation????????? [#permalink] New post 21 Mar 2011, 16:21
Great explanation bagrettin !
Thanks to gmat1220 also.
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Re: line equation????????? [#permalink] New post 24 Oct 2011, 04:45
The explanation is indeed splendid I was searching for some more on equations of line when I found this link on tutorvista which was a great help i think you should also check it out.
Re: line equation?????????   [#permalink] 24 Oct 2011, 04:45
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