JEE Main 2022 (Online) 27th June Morning ShiftQuestion 1
Let the mirror image of the point $(a,b,c)$ with respect to the plane $3x-4y+12z+19=0$ be $(a-6,\beta,\gamma)$. If $a+b+c=5$, then $7\beta-9\gamma$ is equal to $\_\_\_\_\_\_$ .
3D Geometry
Answer$137$
Solution
Let $S=3a-4b+12c+19$. Reflection: $(a,b,c)-\frac{2S}{169}(3,-4,12)$, so $a-\frac{6S}{169}=a-6\Rightarrow S=169$. $\beta=b+8$, $\gamma=c-24$. $3a-4b+12c=150$ with $a=5-b-c$: $15-7b+9c=150\Rightarrow7b-9c=-135$. $7\beta-9\gamma=7b-9c+56+216=-135+272=137$.