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A curve is represented by the equation x^2y^3 = k^3

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A curve is represented by the equation x^2y^3 = k^3  [#permalink]

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New post 27 Aug 2018, 03:35
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39% (01:02) correct 61% (01:25) wrong based on 80 sessions

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A curve is represented by the equation \(x^2y^3 = k^3\), where k < 0. At how many points does the line, y = -a, where a is an integer, intersects this curve?

    A. 0
    B. 1
    C. 2
    D. 4
    E. Cannot be determined

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Re: A curve is represented by the equation x^2y^3 = k^3  [#permalink]

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New post 27 Aug 2018, 04:22
Guess: Since the value of y is always negative, if a is +ve then it cuts the graph at 2 points and if a is -ve then 0 points. Hence cannot be determined.
E??

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A curve is represented by the equation x^2y^3 = k^3  [#permalink]

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New post 27 Aug 2018, 05:54
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can someone sketch solution ? :-) its like chinese to me reading only text without visualization :)
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Re: A curve is represented by the equation x^2y^3 = k^3  [#permalink]

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New post 27 Aug 2018, 06:10
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dave13 wrote:
can someone sketch solution ? :-) its like chinese to me reading only text without visualization :)


This graph looks familiar. Oh wait!!!

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Re: A curve is represented by the equation x^2y^3 = k^3  [#permalink]

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New post 27 Aug 2018, 06:18
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EgmatQuantExpert wrote:
A curve is represented by the equation \(x^2y^3 = k^3\), where k < 0. At how many points does the line, y = -a, where a is an integer, intersects this curve?

    A. 0
    B. 1
    C. 2
    D. 4
    E. Cannot be determined

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IMHO answer is E
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A curve is represented by the equation x^2y^3 = k^3  [#permalink]

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New post 27 Aug 2018, 06:19
1
EgmatQuantExpert wrote:
A curve is represented by the equation \(x^2y^3 = k^3\), where k < 0. At how many points does the line, y = -a, where a is an integer, intersects this curve?

    A. 0
    B. 1
    C. 2
    D. 4
    E. Cannot be determined


The correct answer is (E):

\(\left\{ \begin{gathered}
\,{x^2}{y^3} = {k^3}\,\,\,\left( {k < 0} \right) \hfill \\
y = - a\,\,\,\left( {a\,\,\operatorname{int} } \right) \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,{x^2}{\left( { - a} \right)^3} = {k^3}\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,{x^2} = - {\left( {\frac{k}{a}} \right)^3}\,\,\,\,\,,\,\,\,\,a \ne 0\)

Take (for instance) :

(a,k) = (1,-1) , then we have 2 points (x,y) = (x, -a) of intersection: (-1, -1) and (1, -1)
(a,k) = (-1,-1) , then we have 0 points (x,y) = (x,-a) of intersection, because x^2 = -1 has no solutions (in the real numbers)

The above follows the notations and rationale taught in the GMATH method.
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Re: A curve is represented by the equation x^2y^3 = k^3  [#permalink]

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New post 27 Aug 2018, 06:29
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x^2 is acting as magnitude and y^3 goes from +ve infinite to -ve infinite so it will at least cut y=-a one. so answer is B.
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Re: A curve is represented by the equation x^2y^3 = k^3  [#permalink]

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New post 12 Sep 2018, 09:57
Bunuel can you please explain this?
Re: A curve is represented by the equation x^2y^3 = k^3 &nbs [#permalink] 12 Sep 2018, 09:57
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