JEE 2023 — Mathematics PYQ

Equation of plane intersection of two plane
P₁ + λP₂ = 0

$P_1:2x-y+z=3$, and $P_2:4x-3y+5z+9=0$
$\Rightarrow (2+4\lambda )x-(1+3\lambda )y+(1+5\lambda )z=(3-9\lambda )\dots (i)$

Since given plane parallel to
$\frac{x+1}{-2}=\frac{y+3}{+4}=\frac{z-2}{5}$

$\therefore \vec{b}\cdot \vec{n}=0$
$\Rightarrow -2(2+4\lambda )-4(1+3\lambda )+5(1+5\lambda )=0$
$\Rightarrow \lambda =+\frac{3}{5}$

then plane
$P:11x-7y+10z+6=0$
$\Rightarrow a=11,b=-7,c=10,$
$a+b+c=11-7+10=14$