Question

If $m$ is the slope of one of the lines represented by $a{x}^2+2hxy+b{y}^2=0,$ then $(h+bm{)}^2$ = ______

• A. $h^2+ab$
• B. $h^2-ab$
• C. $(a+b{)}^2$
• D. $(a-b{)}^2$

Answer 1

Answer: (B)

Given that, $a{x}^2+2hxy+b{y}^2=0$...(i)
Which is homogeneous equation representing pair of straight line each of which passing through the origin. Given one slope of line $=m$.
Let another slope of line $=m_1$
Then, the lines are $y=mx$ and $y=m_{1}x$
Now, $(mx-y)\left(m_{1}x-y\right)$
$\Rightarrow m{m}_1{x}^2-m_{1}xy-mxy+y^2$
$\Rightarrow m{m}_1\cdot x^2-\left(m+m_1\right)y\cdot x+y^2$...(ii)
On comparing Eqs. (i) and (ii),
$m+m_1=-\frac{2h}{b}$...(iii)
$m{m}_1=\frac{a}{b}$...(iv)
From Eqs. (iii) and (iv),
$m_1=\left(-\frac{2h}{b}-m\right)$
$\Rightarrow m\left(\frac{-2h}{b}-m\right)=\frac{a}{b}$
$\Rightarrow -\frac{m}{b}(2h+mb)=\frac{a}{b}$
$\Rightarrow -2mh-m^{2}b=a$
$\Rightarrow -2mhb-m^2{b}^2=ab$
$\Rightarrow h^2+2mhb+m^2{b}^2=-ab+h^2$
$\Rightarrow (h+mb{)}^2=h^2-ab$