Question

The length of the major axis and the minor axis of the ellipse $2x^2+3y^2-4x-12y+13=0$ are and respectively.

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. $\frac{2}{\sqrt{3}}$
• D. $\frac{3}{\sqrt{2}}$

Answer 1

Answer: (A), (C)

$\frac{2}{\sqrt{3}}$

The equation can be rewritten as
$2\left(x^2-2x\right)+3\left(y^2-4y\right)+13=0\Rightarrow 2{\left(x-1\right)}^2+3{\left(y-3\right)}^2=1\Rightarrow \frac{{\left(x-1\right)}^2}{(\frac{1}{\sqrt{2}}{)}^2}+\frac{{\left(y-2\right)}^2}{(\frac{1}{\sqrt{3}}{)}^2}=1\mathrm{comparing}\mathrm{it}\mathrm{with}\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1\mathrm{So},\mathrm{the}\mathrm{center}\mathrm{is}h=1,k=2\therefore \mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{major}\mathrm{axis}=2a=\sqrt{2}\mathrm{and}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{minor}\mathrm{axis}=2b=\frac{2}{\sqrt{3}}$