The coordinates of the middle point of the chord intercepted on the line $2x-y+3=0$ by the ellipse $\frac{x^2}{10}+\frac{y^2}{6}=1,$ are

The equation of the ellipse is 

$\frac{x^2}{10}+\frac{y^2}{6}=1\text{ or, }3x^2+5y^2=30$              …(i)

Let$(h, k)$ be the mid-point of the chord intercepted on the line $2x-y+3=0$by the ellipse (i). Then, the equation of the chord is

${S_1}^{ }=S_{11}$

i.e. $3hx+5ky-30=3h^2+5k^2-30$

 $3hx+5ky=3h^2+5k^2$                      …(ii)

The equation of the line is 

$2x-y+3=0$                                         …(iii)

Clearly, (ii) and (iii) represent the same line. 

$\begin{array}{l}\therefore \frac{3h}{2}=-5k=\frac{3h^2+5k^2}{-3}=\lambda \text{ (say) }\\ \Rightarrow h=\frac{2\lambda }{3},k=\frac{-\lambda }{5}\text{ and }3h^2+5k^2=-3\lambda \end{array}$

$\begin{array}{l}\Rightarrow 3\left(\frac{4{\lambda }^2}{9}\right)+5\left(\frac{{\lambda }^2}{25}\right)=-3\lambda \\ \Rightarrow \frac{4{\lambda }^2}{3}+\frac{{\lambda }^2}{5}=-3\lambda \Rightarrow 23{\lambda }^2=-45\lambda \Rightarrow \lambda =0,\lambda =\frac{-45}{23}\end{array}$

But, $\lambda =0$ is not possible. So, $\lambda =\frac{-45}{23}$

Now, $h=\frac{2\lambda }{3},k=\frac{-\lambda }{5}\Rightarrow h=-\frac{30}{23},k=\frac{9}{23}$

Hence, the required coordinates of the mid-point are

$(\frac{-30}{23},\frac{9}{23})$