# Major and minor axes of an ellipse whose center is not at $(0,0)$

Find the equations of major and minor axis of an ellipse $$21x^2-6xy+29y^2+6x-58y-151=0$$ and also eccentricity of an ellipse.

What I tried. Let $$S = 21x^2-6xy+29y^2+6x-58y-151$$

For coordinate of center $$\displaystyle \dfrac{dS}{dx}=0$$ and $$\displaystyle \dfrac{dS}{dy}=0$$. Therefore $$42x-6y+6=0$$ and $$58y-6x-58=0$$

Solving it i have the center $$(0,1)$$.

How do i find the axes?

• What have you tried? Did you try to translate the axes $X=x-0$, $Y=y-1$? – Chrystomath Mar 8 '19 at 11:00

Yes, the center is $$(x,y)=(0,1)$$, so after letting $$X=x$$ and $$Y=y-1$$, the equation of the ellipse can be written as $$21X^2-6XY+29Y^2-180=0.$$ Now, the center is $$(X,Y)=(0,0)$$ and in order to find the axes we need the eigenvectors of the matrix $$\begin{pmatrix} 21 & -3 \\ -3 & 29 \end{pmatrix}$$ whose characteristic equation is $$(21-z)(29-z)-(-3)^2=(z-20)(z-30)=0.$$ For the eigenvalue $$\lambda=20$$, the eigenspace is generated by the vector $$(3,1)^T$$ and therefore one of the axes is $$y-1=Y/1=X/3=x/3$$, i.e. $$y=\frac{x}{3}+1$$.
As regards the eccentricity, find semi-axis's lengths $$a$$, $$b$$ and recall the definition.
Let $$f(x,y)=x^2+(y-1)^2$$ $$\max_{S(x,y)=0}{f}=9,\quad \min_{S(x,y)=0}{f}=6$$ Then semi-major and semi-minor axes are $$a=3,\quad b=\sqrt{6}.$$
• I did not understand How can i find $a$ and $b$ explain me please – jacky Mar 8 '19 at 11:28