Question

Find the centre of the ellipse whose equation is $x^2+2xy+2y^2-1=0$. Also, find the length of the major and minor axes.

• A. Centre: $(1,1)$
• B. Length of major - axis $=\frac{2\sqrt{2}}{\sqrt{3-\sqrt{5}}}$
• C. Length of minor - axis $=\frac{2\sqrt{2}}{\sqrt{3+\sqrt{5}}}$
• D. Centre: $(0,0)$

Answer 1

Answer: (B), (C), (D)

Centre: $(0,0)$

Given: $x^2+2xy+2y^2-1=0$

To Find: $\left(i\right)$ Centre of the ellipse $\left(ii\right)$ Length of major and minor axes

Step-$1:$ Recall the standard equation of conic

Step-$2:$ Recall the method od partial differentiation and find the centre

Step-$3:$ Assume parametric coordinates of any point on ellipse and find major and minor axes

$\left(i\right)$ Centre of the ellipse

The general $2^{nd}$ degree equation of a conic is $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

The centre of this equation can be found using partial differentiation.

$\frac{\delta f(x,y)}{\delta x}=2x+2y=0$

$\frac{\delta f(x,y)}{\delta y}=2x+4y=0$

On solving the two equations we get $x=0$ and $y=0$

So, the centre is $(0,0)$.

$\left(ii\right)$ Length of major and minor axes

Let $P(r\cos \theta ,r\sin \theta )$ be any point on the given ellipse.

$r=$ Maximum distance of point $P$ from the centre $(0,0)$

$(r\cos \theta {)}^2+2(r\sin \theta {)}^2+2r\cos \theta r\sin \theta -1=0$

$\Rightarrow r^2{\cos }^2\theta +2r^2{\sin }^2\theta +2r^2\cos \theta \sin \theta -1=0$

$\Rightarrow r^2({\cos }^2\theta +{\sin }^2\theta )+r^2{\sin }^2\theta +2r^2\cos \theta \sin \theta -1=0$

$\Rightarrow r^2+r^2(\frac{1-\cos 2\theta }{2})+2r^2\sin 2\theta -2=0$

$\Rightarrow 3r^2+r^2(2\sin 2\theta -\cos 2\theta )=2$

$\Rightarrow r^2=\frac{2}{3+2\sin 2\theta -\cos 2\theta }$

Maximum length of $r$ = Length of semi - major axis

Minimum length of $r$ = Length of semi - minor axis

$-\sqrt{2^2+(-1{)}^2}\le 2\sin 2\theta -\cos 2\theta \le \sqrt{2^2+(-1{)}^2}$

$-\sqrt{5}\le 2\sin 2\theta -\cos 2\theta \le \sqrt{5}$

$r_{max}^2=\frac{2}{3-\sqrt{5}}\text{and}{r}_{min}^2=\frac{2}{3+\sqrt{5}}$

$r_{max}=\frac{\sqrt{2}}{\sqrt{3-\sqrt{5}}}\text{and}{r}_{min}=\frac{\sqrt{2}}{\sqrt{3+\sqrt{5}}}$

Length of major - axis $=2r_{max}=\frac{2\sqrt{2}}{\sqrt{3-\sqrt{5}}}$

Length of minor - axis $=2r_{min}=\frac{2\sqrt{2}}{\sqrt{3+\sqrt{5}}}$