question_answer

If the slope of one of the lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] is the square of the other, Then \[\frac{a+b}{h}+\frac{8{{h}^{2}}}{ab}=\]

A) 4

B) 6

C) 8

D) None of these

Correct Answer: B

Solution :

[b] Let m and \[{{m}^{2}}\] be the slopes of the lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0.\] Then, \[m+{{m}^{2}}=-\frac{2h}{b}\] and \[m{{m}^{2}}=\frac{a}{b}\] or \[{{m}^{3}}=\frac{a}{b}\] \[\therefore \,\,\,\,\,\,{{(m+{{m}^{2}})}^{3}}={{\left( -\frac{2{{h}^{2}}}{b} \right)}^{2}}\]\[\therefore {{m}^{3}}+{{m}^{6}}+3m{{m}^{2}}(m+{{m}^{2}})=-\frac{8{{h}^{3}}}{{{b}^{3}}}\] \[\therefore \frac{a}{{{b}^{2}}}(a+b)+\frac{8{{h}^{3}}}{{{b}^{3}}}=\frac{6ah}{{{b}^{2}}}\therefore \frac{a+b}{h}+\frac{8{{h}^{2}}}{ab}=6\] These are the set of parallel lines and the distance between parallel lines are equal. So, the figure is a rhombus.