According to ques,
Coordinates are \[A\text{ }\left( 2,\text{ }3,\text{ }4 \right)\text{ }and\text{ }B\text{ }\left( 4,\text{ }5,\text{ }8 \right)\] Again,
Coordinates of the mid-point C are \[\left( 2+4/2,\text{ }3+5/2,\text{ }4+8/2 \right)\text{ }=\text{ }\left( 3,\text{ }4,\text{ }6 \right)\]
Since the direction ratios of the normal to the plane = direction ratios of AB,
Hence,
\[=\text{ }4\text{ }\text{ }2,\text{ }5\text{ }\text{ }3,\text{ }8\text{ }\text{ }4\text{ }=\text{ }\left( 2,\text{ }2,\text{ }4 \right)\]
Therefore,
Equation of the plane :
\[a\left( x\text{ }\text{ }{{x}_{1}} \right)\text{ }+\text{ }b\left( y\text{ }\text{ }{{y}_{1}} \right)\text{ }+\text{ }c\left( z\text{ }\text{ }{{z}_{1}} \right)\text{ }=\text{ }0\]
Or,
\[2\left( x\text{ }\text{ }3 \right)\text{ }+\text{ }2\left( y\text{ }\text{ }4 \right)\text{ }+\text{ }4\left( z\text{ }\text{ }6 \right)\text{ }=\text{ }0\]
Or,
\[2x\text{ }\text{ }6\text{ }+\text{ }2y\text{ }\text{ }8\text{ }+\text{ }4z\text{ }\text{ }24\text{ }=\text{ }0\]
Or,
\[2x\text{ }+\text{ }2y\text{ }+\text{ }4z\text{ }=\text{ }38\]
Or,
\[x\text{ }+\text{ }y\text{ }+\text{ }2z\text{ }=\text{ }19\]
Required equation of plane :
\[x\text{ }+\text{ }y\text{ }+\text{ }2z\text{ }=\text{ }19\]
or