A) \[hg+ab=0\]
B) \[ah+bg=0.\]
C) \[{{h}^{2}}-ab=0\]
D) \[ag+bh=0.\]
Correct Answer: A
Solution :
Since, each pair of line bisect the angle between the other. So, the equation of bisectors of the angles between the line given by \[a{{x}^{2}}+2hxy-a{{y}^{2}}=0\]is \[b{{x}^{2}}+2gxy-b{{y}^{2}}=0\] ...(i) But the equation of bisector of the angle between the line given by \[a{{x}^{2}}+2hxy-a{{y}^{2}}=0\]is \[\frac{{{x}^{2}}-{{y}^{2}}}{a-(-a)}=\frac{xy}{h}\] \[\Rightarrow \] \[h{{x}^{2}}-2axy-h{{y}^{2}}=0\] ...(ii) Since, lines (i) and (ii) represents the same pair of straight lines, therefore \[\frac{b}{h}=\frac{g}{-a}=\frac{-b}{-h}\] \[\Rightarrow \] \[-ab=gh\] \[\Rightarrow \] \[ab+gh=0\]
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