QJEEM24Ma5151: JEE Main 2024 Mathematics - Class 11 JEEM24Ma5151 Solution: If is the locus of a point, which moves such that it is always…

Cocus of point $\mathrm{P}(\mathrm{x}, \mathrm{y})$ whose distance from Gives $X+2 y+7=0$ & $2 x-y+8=0$ are equal is $\frac{x+2 y+7}{\sqrt{5}}= \pm \frac{2 x-y+8}{\sqrt{5}}$

$(x+2 y+7)^2-(2 x-y+8)^2=0$

Combined equation of lines

$\begin{aligned}& (x-3 y+1)(3 x+y+15)=0 \\& 3 x^2-3 y^2-8 x y+18 x-44 y+15=0 \\& x^2-y^2-\frac{8}{3} x y+6 x-\frac{44}{3} y+5=0 \\& x^2-y^2+2 h x y+2 g x 2+2 f y+c=0 \\& h=\frac{4}{3}, g=3, f=-\frac{22}{3}, c=5 \\& g+c+h-f=3+5-\frac{4}{3}+\frac{22}{3}=8+6=14\end{aligned}$