Question

Let $ABC$ be a right angled triangle with angle at $A(2,1)$ being ${90}^o$. If the equation of pair of the line $BC$ be $2x+y=3$ then the joining equation of lines $AB$ and $AC$ is $3x^2-3y^2+8xy-20x-10y+25=0$

Answer 1

Let $m$ be the slope of any of the lines $AB$ and $BC$ each inclined at ${45}^o$ to $2x+y=3(slope-2).$
$\therefore \tan (\pm {45}^o)=\frac{m+2}{1-2m}=\pm 1$
$\therefore m+2=1-2m$
and $m+2=-(1-2m)$
$\therefore m=-1/3,m=3$
Hence, the two lines are
$y-1=3(x-2)$ and $y-1=-\frac{1}{3}(x-2)$
or $3x-y-4=0$ and $x+3y-5=0$
Their combined equation is
$(3x-y-5)(x+3y-5)=0$
or $3x^2-3y^2+8xy-20x-10y+25=0$