Let the line passing through $\left(-3, 4\right)$ intersect the coordinate axes at $A$ & $B$.

By mid-point formula, co-ordinates of $P\left(\frac{\alpha +0}{2}, \frac{0+\beta }{2}\right)$.

Comparing with given co-ordinate values, we get

$\frac{\alpha +0}{2}=-3\Rightarrow \alpha =-6$

and $\frac{\beta + 0}{2}=4\Rightarrow \beta =8$

Hence, equation of line is $\frac{x}{-6}+\frac{y}{8}=1$

$\Rightarrow 4x-3y+24=0$