The locus of mid-points of the line segments

General point on $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ is $A(2 \cos \theta, 3 \sin \theta)$

given $\mathrm{B}(-3,-5)$

midpoint $C\left(\frac{2 \cos \theta-3}{2}, \frac{3 \sin \theta-5}{2}\right)$

$\mathrm{h}=\frac{2 \cos \theta-3}{2} ; \mathrm{k}=\frac{3 \sin \theta-5}{2}$

$\Rightarrow\left(\frac{2 \mathrm{~h}+3}{2}\right)^{2}+\left(\frac{2 \mathrm{k}+5}{3}\right)^{2}=1$

$\Rightarrow 36 x^{2}+16 y^{2}+108 x+80 y+145=0$