Source: e-GMAT
A curve is represented by the equation x^2 y^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 OA is E.
A curve is represented by the equation x^2 y^3 = k^3, where
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- Jay@ManhattanReview
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Let's plug-in y = - a in the equation x^2*y^3 = k^3BTGmoderatorLU wrote:Source: e-GMAT
A curve is represented by the equation x^2 y^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 OA is E.
-x^2*a^3 = -|k^3|; we are given that k < 0
x^2*a^3 = |k^3|
x^2 = |k^3| / a^3
There are three cases...
1. If a > 0 then x^2 = |k^3| / |a^3| => x = ±√(|k/a|^3. There are two points.
2. If a = 0 then x^2 = |k^3| / |0^3| is indeterminable.
3. If a ≤ 0 then x^2 = |k^3| / -|a^3| => x^2 = -[|k^3| / |a^3|]; squareroot of a negative number is not possible. Indeterminable.
The correct answer: E
Hope this helps!
-Jay
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- fskilnik@GMATH
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The correct answer is (E):BTGmoderatorLU wrote:Source: e-GMAT
A curve is represented by the equation x^2 y^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
\[\left\{ \begin{gathered}
\,{x^2}{y^3} = {k^3}\,\,\,\left( {k < 0} \right) \hfill \\
y = - a\,\,\,\left( {a\,\,\operatorname{int} } \right) \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,{x^2}{\left( { - a} \right)^3} = {k^3}\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,{x^2} = - {\left( {\frac{k}{a}} \right)^3}\,\,\,\,\,,\,\,\,\,a \ne 0\]
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)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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