A) g = 4, f = 9, c = 0
B) g = 2, f = 3, c = 1
C) g = 2, f = 3, c is any number
D) g = 4, f = 9, c > 1
Correct Answer: C
Solution :
The lines are parallel, if \[{{h}^{2}}=ab,\,a{{f}^{2}}=b{{g}^{2}}\] or \[\frac{a}{h}=\frac{h}{b}=\frac{g}{f}\Rightarrow 4{{f}^{2}}=9{{g}^{2}}\] \[\Rightarrow f=\frac{3}{2}g\Rightarrow g=2,\ \ f=3\] (let) Now \[abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0\] \[\Rightarrow 4\times 9\times c+2\times 3\times 2\times 6-4{{(3)}^{2}}-9{{(2)}^{2}}-c{{(6)}^{2}}=0\] \[\Rightarrow \]c is any number.
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