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Grockit - Coordinate Geometry Q

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Absco: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Mon Jun 27, 2011 1:59 pm; Location: London

by Absco » Tue Apr 17, 2012 5:12 am

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

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Source: Beat The GMAT — Data Sufficiency | Tue Apr 17, 2012 5:12 am

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Tue Apr 17, 2012 5:24 am

Absco wrote: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

Slope of line y = px + q is p and slope of line y = mx + n is m.
(1) m = p + 2 implies there is just one equation and 2 variables. Until we know the values of m and p, we cannot say whether reflection of y = px + q is parallel to the line y = mx + n. So, (1) is NOT SUFFICIENT.

(2) m = 3p again implies there is just one equation and 2 variables. Until we know the values of m and p, we cannot say whether reflection of y = px + q is parallel to the line y = mx + n. So, (2) is NOT SUFFICIENT.

Combining (1) and (2), 3p = p + 2 implies p = 1 and m = 3. So, now we can find the angle that each of the two lines make with x -axis. Hence, combining the statements is SUFFICIENT to answer the question.

The correct answer is C.

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aneesh.kg: Master | Next Rank: 500 Posts; Posts: 385; Joined: Mon Apr 16, 2012 8:40 amFollowed by: 29 members; Location: Pune, India; Thanked: 186 times

by aneesh.kg » Tue Apr 17, 2012 5:42 am

Let's try to find the slope of the reflected line.

The point of intersection of the lines y = px + q and y = x is (q/1 - p, q/1 - p)
Let's take a simple point on the line y = px + q as (0 , q). The reflection of (0 , q) about the line y = x will be (q , 0).

The reflected line must be passing through (q/1 - p, q/1 - p) and (q , 0).

Thus, Slope of the line = [((q/1 - p) - q)/ (q/1 - p) - (0))] = 1/p

So, in order to find if this line is parallel to y = mx + c or not, we need to see if 1/p = m or not. Or, Is mp = 1 ?

Both the Statements independently provide infinite solutions for m and p, including the ones that satisfy mp = 1 and the ones that don't, and are thus INSUFFICIENT INDEPENDENTLY.

We need to combine both the statements to get a unique value for m and p and thus answer the above question.

Therefore, (C) is the answer.

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bryan88: Master | Next Rank: 500 Posts; Posts: 120; Joined: Mon Nov 07, 2011 10:54 pm; Location: Delhi; Thanked: 5 times

by bryan88 » Tue Apr 17, 2012 7:35 am

We need to check whether m=p; doesnt B do that for us?
Am i making a blunder here?

Anurag@Gurome wrote:
Absco wrote: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

Slope of line y = px + q is p and slope of line y = mx + n is m.
(1) m = p + 2 implies there is just one equation and 2 variables. Until we know the values of m and p, we cannot say whether reflection of y = px + q is parallel to the line y = mx + n. So, (1) is NOT SUFFICIENT.

(2) m = 3p again implies there is just one equation and 2 variables. Until we know the values of m and p, we cannot say whether reflection of y = px + q is parallel to the line y = mx + n. So, (2) is NOT SUFFICIENT.

Combining (1) and (2), 3p = p + 2 implies p = 1 and m = 3. So, now we can find the angle that each of the two lines make with x -axis. Hence, combining the statements is SUFFICIENT to answer the question.

The correct answer is C.

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aneesh.kg: Master | Next Rank: 500 Posts; Posts: 385; Joined: Mon Apr 16, 2012 8:40 amFollowed by: 29 members; Location: Pune, India; Thanked: 186 times

by aneesh.kg » Tue Apr 17, 2012 8:07 am

We don't need to check if m = p. We have to check if m = (slope of the reflected line).

If you read my explanation above, the question reduces to 'm = 1/p ?'

Aneesh Bangia
GMAT Math Coach
[email protected]

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ollapodrida: Senior | Next Rank: 100 Posts; Posts: 37; Joined: Thu Oct 06, 2011 2:49 pm; Thanked: 4 times

by ollapodrida » Wed Apr 18, 2012 1:41 am

aneesh.kg wrote:Let's try to find the slope of the reflected line.

Thus, Slope of the line = [((q/1 - p) - q)/ (q/1 - p) - (0))] = 1/p

So, in order to find if this line is parallel to y = mx + c or not, we need to see if 1/p = m or not. Or, Is mp = 1 ?

You are assuming here that the two lines with slopes m and p must be perpendicular in order to be parallel after reflecting one of the lines about the line y=x. I'm not sure if this assumption is valid.

I think the two lines only need be equal angles away from y=x in order for the reflected line to be parallel with the other line. The picture posted by Anurag might shed some light on what I'm saying -- that m*p need not equal -1?, although in this case, the problem can be solved if mp = any constant, as long as m and p are known, as shown in Anurag's post.

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subhakam: Senior | Next Rank: 100 Posts; Posts: 38; Joined: Mon Jun 21, 2010 12:37 pm

by subhakam » Sun Mar 03, 2013 12:51 pm

Anurag@Gurome wrote:
Absco wrote: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

Slope of line y = px + q is p and slope of line y = mx + n is m.
(1) m = p + 2 implies there is just one equation and 2 variables. Until we know the values of m and p, we cannot say whether reflection of y = px + q is parallel to the line y = mx + n. So, (1) is NOT SUFFICIENT.

(2) m = 3p again implies there is just one equation and 2 variables. Until we know the values of m and p, we cannot say whether reflection of y = px + q is parallel to the line y = mx + n. So, (2) is NOT SUFFICIENT.

Combining (1) and (2), 3p = p + 2 implies p = 1 and m = 3. So, now we can find the angle that each of the two lines make with x -axis. Hence, combining the statements is SUFFICIENT to answer the question.

The correct answer is C.

Hello Anurag - I did not understand the question or the solution - What does the term "reflection mean" ? I was assuming that the line y=x is actually perpendicular to y=px+q therefore slope of y=x; M" = 1 and slope of y=px+q; p= -1 ?? And thus question is asking if this line y=px+q is parallel to line y=mx+n; i.e. m= -1 ?

I totally misunderstood the question. Appreciate if you could explain it to me
Many thanks
Subhakam

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by Anurag@Gurome » Sun Mar 03, 2013 9:43 pm

subhakam wrote:Hello Anurag - I did not understand the question or the solution - What does the term "reflection mean" ?
...
I totally misunderstood the question. Appreciate if you could explain it to me

In coordinate geometry, reflection over or with respect to a line means as if there is a invisible mirror placed on the line. For example, in the following figure P' is the reflection of P with respect to the line l, meaning as if there is an invisible mirror along the line l on which P' is the reflection of P.

Now according to the law of reflection,

1. Shortest distance of P from l = Shortest distance of P' from l
2. The straight line joining P and P' will be perpendicular to the line l

Now in our case a straight line (y = px + q) is reflected with respect to another straight line, y = x. This means we have to consider that an invisible mirror is placed along y = x and y = px + q is reflected on it. Refer to the following figures,

In the first figure, the black line is y = x and assume that the blue line is y = px + q. In the second figure, the red line is the reflection of the blue line over the black line.

The concept can be similarly extended to more complex shapes like triangle, circles etc also.

I hope this helps you to get an idea of what reflection means.

There are formulas to find out what will be the equation of a reflected line over an axis or another line etc. However, we can solve this particular question without any such formula. We just need to understand that we need to know whether slopes of the reflection of y = px + q is equal to m or not. This in turn means we need to know the slope of y = px + q, i.e. p and m.

Now, you can follow my reply or aneesh.kg's reply for a more detailed mathematical approach.

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