Question

The equation of the straight line passing through the point of intersection of the straight lines$\frac{x}{a}=\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$ and having infinite slopes is

Answer 1

Let $L_1:\frac{x}{a}+\frac{y}{b}-1=0$ & $L_2=\frac{x}{b}+\frac{y}{a}-1=0$

the required line would be of the form $L_1-\lambda L_2=0$

$\Rightarrow (\frac{x}{a}+\frac{y}{b}-1)-\lambda (\frac{x}{b}+\frac{y}{a}-1)=0$

$\Rightarrow x(\frac{1}{a}-\frac{\lambda }{b})+y(\frac{1}{b}-\frac{\lambda }{a})-1+\lambda =0$

As slope = $\infty \Rightarrow$ line is parallel to y-axis

$\Rightarrow$ coeff. of $y=0\Rightarrow \frac{\lambda }{a}=\frac{1}{b}\Rightarrow \lambda =\frac{a}{b}$

$\Rightarrow E{q}^n$ of line : $x(\frac{1}{a}-\frac{a}{b^2})+y(\frac{1}{b}-\frac{1}{b})-1+\frac{a}{b}=0$

$\Rightarrow x\left(\frac{b^2-a^2}{a{b}^2}\right)+\frac{a}{b}-1=0$