Question

Two parallel straight lines at an angle of
${135}^o$ to the axis of $x$ meet the $x-$axis at the points
$A,B$ and $y-$axis at the points $C$ and $D$
respectively. Find the equation of the locus of point of intersection of $AD$ and $BC$.

Answer 1

$\tan {135}^o=-1=m$. Let the two lines be $x+y=a,x+y=b$
They meet the axis of $x$ in $A,B$, $y-$axis in $C,D$.
$\therefore A(a,0)$, $B(b,0)$, $C(0,a)$, $D(0,b)$
By intercepts from equation of $AD$ and $BC$ are
$AD,\frac{x}{a}+\frac{y}{b}=1;BC,\frac{x}{b}+\frac{y}{a}=1$
We have to eliminate the variable $a,b$ in order to find the locus of their point of intersection. Subtracting we get
$x(\frac{1}{a}-\frac{1}{b})-y(\frac{1}{a}-\frac{1}{b})=0$ $\therefore x-y=0$
is the required locus