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The straight lines given by the equation 2x^2=2y^2-3xy are:

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GMATH Teacher
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Joined: 12 Oct 2010
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The straight lines given by the equation 2x^2=2y^2-3xy are:  [#permalink]

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21 Mar 2019, 13:19
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55% (hard)

Question Stats:

56% (02:28) correct 44% (02:07) wrong based on 27 sessions

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GMATH practice exercise (Quant Class 20)

The straight lines given by the equation $$2{x^2} = 2{y^2} - 3xy$$ are:

(A) parallel
(B) intersecting and they form a 30-degrees angle
(C) intersecting and they form a 45-degrees angle
(D) intersecting and they form a 60-degrees angle
(E) intersecting and they form a 90-degrees angle

_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Re: The straight lines given by the equation 2x^2=2y^2-3xy are:  [#permalink]

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21 Mar 2019, 14:47
1
fskilnik wrote:
GMATH practice exercise (Quant Class 20)

The straight lines given by the equation $$2{x^2} = 2{y^2} - 3xy$$ are:

(A) parallel
(B) intersecting and they form a 30-degrees angle
(C) intersecting and they form a 45-degrees angle
(D) intersecting and they form a 60-degrees angle
(E) intersecting and they form a 90-degrees angle

$$?\,\,:\,\,{\rm{lines}}\,\,{\rm{relative}}\,\,{\rm{position}}$$

$$2{x^2} = 2{y^2} - 3xy\,\,\,\, \Leftrightarrow \,\,\,\,2{x^2} - xy = 2{y^2} - 4xy\,\,\,\, \Leftrightarrow$$

$$\Leftrightarrow \,\,\,\,x\left( {2x - y} \right) = - 2y\left( { - y + 2x} \right)\,\,\,\, \Leftrightarrow \,\,\,\,\left( {2x - y} \right)\left( {x + 2y} \right) = 0$$

$$\left( {2x - y} \right)\left( {x + 2y} \right) = 0\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ \matrix{ \,2x - y = 0 \hfill \cr \,\,{\rm{or}} \hfill \cr \,x + 2y = 0 \hfill \cr} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ \matrix{ \,y = 2x\,\,\,\,\left( {{\rm{slope}} = 2} \right) \hfill \cr \,\,{\rm{or}} \hfill \cr \,y = - {x \over 2}\,\,\,\,\left( {{\rm{slope}} = - {1 \over 2}} \right) \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{E}} \right)$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: The straight lines given by the equation 2x^2=2y^2-3xy are:  [#permalink]

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21 Mar 2019, 17:44
fskilnik wrote:
fskilnik wrote:
GMATH practice exercise (Quant Class 20)

The straight lines given by the equation $$2{x^2} = 2{y^2} - 3xy$$ are:

(A) parallel
(B) intersecting and they form a 30-degrees angle
(C) intersecting and they form a 45-degrees angle
(D) intersecting and they form a 60-degrees angle
(E) intersecting and they form a 90-degrees angle

$$?\,\,:\,\,{\rm{lines}}\,\,{\rm{relative}}\,\,{\rm{position}}$$

$$2{x^2} = 2{y^2} - 3xy\,\,\,\, \Leftrightarrow \,\,\,\,2{x^2} - xy = 2{y^2} - 4xy\,\,\,\, \Leftrightarrow$$

$$\Leftrightarrow \,\,\,\,x\left( {2x - y} \right) = - 2y\left( { - y + 2x} \right)\,\,\,\, \Leftrightarrow \,\,\,\,\left( {2x - y} \right)\left( {x + 2y} \right) = 0$$

$$\left( {2x - y} \right)\left( {x + 2y} \right) = 0\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ \matrix{ \,2x - y = 0 \hfill \cr \,\,{\rm{or}} \hfill \cr \,x + 2y = 0 \hfill \cr} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ \matrix{ \,y = 2x\,\,\,\,\left( {{\rm{slope}} = 2} \right) \hfill \cr \,\,{\rm{or}} \hfill \cr \,y = - {x \over 2}\,\,\,\,\left( {{\rm{slope}} = - {1 \over 2}} \right) \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{E}} \right)$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Hi,
Is not it the equation above second degree that does present a line? As I know, an equation of line is first degree. Can you help please with more details?
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
The straight lines given by the equation 2x^2=2y^2-3xy are:  [#permalink]

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21 Mar 2019, 18:03
Mo2men wrote:

Hi,
Is not it the equation above second degree that does present a line? As I know, an equation of line is first degree. Can you help please with more details?

Hi Mo2men,

Thank you for your interest in our problem and in my solution.

I will not go into many details about the relationship between lines and first-degree equations, because all that it is needed to know (in GMAT´s scope) is contained in our course.

On the other hand, please notice that the set of points (x,y) in the rectangular coordinate system, associated with the solution set of the second-degree equation presented in the question stem, is exactly the reunion of TWO intersecting lines, not just one line.

In other words, your intuition/knowledge is not refuted: the second-degree equation presented in the question stem is NOT related to a SINGLE straight line!

Regards and success in your studies,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
The straight lines given by the equation 2x^2=2y^2-3xy are:   [#permalink] 21 Mar 2019, 18:03
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