Question

If one of the slopes of the pair of lines $a{x}^2+2hxy+b{y}^2=0$ is n times the other then

• A. $4(n+1{)}^{2}ab=nab$
• B. $4h^2(n+1{)}^{2}ab$
• C. $4n{h}^2=(n+1{)}^{2}ab$
• D. $4ab=(n+1{)}^{2}h$

Answer 1

The correct option is C $4n{h}^2=(n+1{)}^{2}ab$

$a{x}^2+2hxy+b{y}^2=0$

$\Rightarrow b{\left(\frac{y}{x}\right)}^2+2h\left(\frac{y}{x}\right)+a=0$

Since the slope is given by $\frac{y}{x}$, sum of the slopes is given by $\frac{-2h}{b}$ and their product is given by $\frac{a}{b}$

Since one slope is n times the other, $m(n+1)=\frac{-2h}{b}$ and $n{m}^2=\frac{a}{b}$ where m is the smaller slope.

$\Rightarrow {\left(\frac{-2h}{b(n+1)}\right)}^2=\frac{a}{bn}$

$\Rightarrow \frac{4h^2}{b^2(n+1{)}^2}=\frac{a}{nb}$

$\Rightarrow 4n{h}^2=(n+1{)}^{2}ab$