For the ellipse  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ (where a > b), if the locus of centroid of the triangle formed by the centre and the points of intersection of chord with ellipse which subtends right angle at the origin is $4a^2{b}^2\left(x^2{b}^4+y^2{a}^4\right)=K\left(a^2+b^2\right){\left(x^2{b}^2+y^2{a}^2\right)}^2$  , the value of 4K _________

Centroid

$\begin{array}{l}G\left(\frac{x_1+x_2}{3},\frac{y_1+y_2}{3}\right)\left(\alpha ,\beta \right)=\left(\frac{x_1+x_2}{3},\frac{y_1+y_2}{3}\right)\\ x_1+x_2=3\alpha ,y_1+y_2=3\beta \left(x_1,y_1\right)\&\left(x_2,y_2\right)\\ arepoint7\\ Ellipse\Rightarrow \frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=1\_\_\_\_\left(1\right)\\ \frac{x_2^2}{a^2}+\frac{y_2^2}{b^2}=1\_\_\_\_\_\left(2\right)\\ \left(1\right)-\left(2\right)\Rightarrow \frac{\left(x_1-x_2\right)\left(x_1+x_2\right)}{a^2}+\frac{\left(y_1-y_2\right)\left(y_1+y_2\right)}{b^2}=0\\ \Rightarrow \frac{\left(x_1-x_2\right)\left(3\alpha \right)}{a^2}+\frac{\left(y_1-y_2\right)\left(3\beta \right)}{b^2}=0\\ \Rightarrow slopeoflineAB=\frac{y_2-y_1}{x_2-x_1}\\ m=-\frac{b^2\alpha }{a^2\beta }\\ ConsiderlineABasy=mx+c\\ \left(x_1,y_1\right)and\left(x_2,y_2\right)liesonit\\ \left.\begin{array}{l}{y}_1=m{x}_1+c\\ y_2=m{n}_2+c\end{array}\right]\Rightarrow 3\beta =m\left(3\alpha \right)+2c\\ \Rightarrow c=\frac{\left(\beta -m\alpha \right)3}{2}\\ Homogenizingellipsewithline\\ \frac{b^2{x}^2+a^2{y}^2}{a^2+b^2}-{\left(\frac{y-mx}{c}\right)}^2=0\\ Rightangleatorigin\\ \Rightarrow \frac{1}{a^2}+\frac{m^2}{c^2}+\frac{1}{b^2}-\frac{1}{c^2}=0\end{array}$