Given equation of plane, $P_1:2x+y-5z=0$and $P_2:3x-y+4z-7=0$

We know that equation of family of planes $P_1$ and $P_2$ is given by,

$P:P_1+\lambda P_2$

$\therefore P:\left(2+3\lambda \right)x+\left(1-\lambda \right)y+\left(-5+4\lambda \right)z-7\lambda =0$

$\because P\perp P_1 \therefore 4+6\lambda +1-\lambda +25-20\lambda =0\Rightarrow \lambda =2$

So, equation of plane $P:8x-y+3z-14=0$

It passes through the point $\left(1,0,2\right)$