Mathematics - Application of Derivative Question with Solution | TestHub

The equation of ellipse is $$\begin{array}{l}3x^2+2y^2+6x-8y+5=0\\ \Rightarrow 3\left(x^2+2x\right)+2\left(y^2-4y\right)+5=0\\ \Rightarrow 3\left(x^2+2x+1\right)+2\left(y^2-4y+4\right)\\ +5-3-8=0\end{array}$$ $$\begin{array}{l}\Rightarrow 3(x+1{)}^2+2(y-2{)}^2=6\\ \Rightarrow \frac{(x+1{)}^2}{2}+\frac{(y-2{)}^2}{3}=1\end{array}$$ Comparing with $$\begin{array}{l}\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1,\text{ we get }\\ h=-1,k=2,a^2=2,b^2=3\\ \text{ Here, centre }(h,k)=(-1,2)\\ \text{ And using }{a}^2=b^2\left(1-e^2\right)\\ 2=3\left(1-e^2\right)\Rightarrow e=\frac{1}{\sqrt{3}}\\ \text{ And foci are }(h,k+be)\text{ and }(h,k-be)\\ =(-1,2+1)\text{ and }(-1,2-1)\\ =(-1,3)\text{ and }(-1,1)\end{array}$$