$$Statement-1$$: If −$$2h=a+b$$, then one line of the pair of lines $$ax2+2hxy+by2=0$$ bisects the angle between the coordinate axes in the positive quadrant.Statement $$2$$: If $$ax+y(2h+a)=0$$ is a factor of $$ax2+2hxy+by2=0$$,then $$b+2h+a=0$$.

Question

$$Statement-1$$: If −$$2h=a+b$$, then one line of the pair of lines $$ax2+2hxy+by2=0$$ bisects the angle between the coordinate axes in the positive quadrant.Statement $$2$$: If $$ax+y(2h+a)=0$$ is a factor of $$ax2+2hxy+by2=0$$,then $$b+2h+a=0$$.

Infinity Learn · Accepted Answer

Put $$2h=$$−(a+b) in $$ax2+2hxy+by2=0$$. Then, $$ax2$$−$$(a+b)xy+by2=0or$$ $$(x$$−$$y)(ax$$−$$by)=0Therefore$$, one of the lines bisects the angle between the coordinates axes in the positive quadrant. Also, putting $$b=$$−$$2h$$−a in ax−by,we have ax−$$by=ax$$−$$($$−$$2h$$−$$a)y=ax+(2h+a)yHence$$, $$ax+(2h+a)y$$  is a factor of $$ax2+2hxy$$.However, statement $$2$$ is not the correct explanation of statement $$1$$.

Statement-1: If $- 2 h = a + b,$ then one line of the pair of lines ${ax}^{2} + 2 hxy + {by}^{2} = 0$ bisects the angle between the coordinate axes in the positive quadrant.
Statement 2: If $ax + y (2 h + a) = 0$ is a factor of ${ax}^{2} + 2 hxy + {by}^{2} = 0,$ then $b + 2 h + a = 0 .$

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Statement-1: If −2h=a+b, then one line of the pair of lines ax2+2hxy+by2=0 bisects the angle between the coordinate axes in the positive quadrant.Statement 2: If ax+y(2h+a)=0 is a factor of ax2+2hxy+by2=0,then b+2h+a=0.

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Statement-1: If $- 2 h = a + b,$ then one line of the pair of lines ${ax}^{2} + 2 hxy + {by}^{2} = 0$ bisects the angle between the coordinate axes in the positive quadrant.
Statement 2: If $ax + y (2 h + a) = 0$ is a factor of ${ax}^{2} + 2 hxy + {by}^{2} = 0,$ then $b + 2 h + a = 0 .$