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16 Nov 2021, 05:15
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In the \(xy\)-plane, line \(k\) intercepts \(x\)-axis at \((a, 0)\) and \(y\)-axis at \((0, b)\). If \(ab ≠ 0\), what is the slope of line \(k\) ?
(1) \(a^2 = b^2\)
(2) \(a^3 = b^3\)
(1) \(a^2 = b^2\)
(2) \(a^3 = b^3\)
16 Nov 2021, 05:15
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Official Solution:
In the xy-plane, line \(k\) intercepts x-axis at \((a, 0)\) and y-axis at \((0, b)\). If \(ab ≠ 0\), what is the slope of line \(k\) ?
The slope of line \(k\) is given by: \(slope=\frac{rise}{run}=\frac{change \ in \ y}{change \ in \ x}\).
(1) \(a^2 = b^2\)
\(a=b\) or \(a=-b\)
If \(a=b\), then the lines passes through \((a, 0)\) and \((0, a)\), so \(slope=\frac{rise}{run}=\frac{0-a}{a-0}=-1\)
If \(a=-b\), then the lines passes through \((a, 0)\) and \((0, -a)\), \(\frac{rise}{run}=\frac{0-(-a)}{a-0}=1\)
Not sufficient.
(2) \(a^3 = b^3\)
\(a=b\). The lines passes through \((a, 0)\) and \((0, a)\), so \(slope=\frac{rise}{run}=\frac{0-a}{a-0}=-1\)
Sufficient.
Answer: B
In the xy-plane, line \(k\) intercepts x-axis at \((a, 0)\) and y-axis at \((0, b)\). If \(ab ≠ 0\), what is the slope of line \(k\) ?
The slope of line \(k\) is given by: \(slope=\frac{rise}{run}=\frac{change \ in \ y}{change \ in \ x}\).
(1) \(a^2 = b^2\)
\(a=b\) or \(a=-b\)
If \(a=b\), then the lines passes through \((a, 0)\) and \((0, a)\), so \(slope=\frac{rise}{run}=\frac{0-a}{a-0}=-1\)
If \(a=-b\), then the lines passes through \((a, 0)\) and \((0, -a)\), \(\frac{rise}{run}=\frac{0-(-a)}{a-0}=1\)
Not sufficient.
(2) \(a^3 = b^3\)
\(a=b\). The lines passes through \((a, 0)\) and \((0, a)\), so \(slope=\frac{rise}{run}=\frac{0-a}{a-0}=-1\)
Sufficient.
Answer: B
