Question

To the lines $ax^2 + 2hxy + by^2 = 0$ , the line $a^2x^2 + 2h(a + b)xy + b^2y^2 = 0$ are

• A. equally inclined
• B. perpendicular
• C. bisector of the angle
• D. None of the above

Answer 1

Answer: (A)

The equation of the bisectors of the angle between the lines given by
$ax^{2}+2\,hxy+by^{2}=0$ is
$\frac{x^{2}-y^{2}}{a-b}=\frac{xy}{h}\,...\left(i\right)$
And the equation of the bisectors of the angle between the lines given by
$a^{2}x^{2}+2h\left(a+b\right)xy+b^{2}y^{2}=0$ is
$\frac{x^{2}-y^{2}}{a^{2}-b^{2}}=\frac{xy}{h\left(a+b\right)}$
$\Rightarrow \frac{x^{2}-y^{2}}{a-b}=\frac{x}{h}\,...\left(ii\right)$
From eqs. $\left(i\right)$ and $\left(ii\right)$, it is clear that both the pair of straight lines have the same bisector, hence, the given two pairs of straight lines are equally inclined.