Question

The equation of the ellipse, whose axes are of lengths $6$ and $2\sqrt{6}$ and their equations are $x-3y+3=0$ and $3x+y-1=0$ respectively, is

• A. $21x^2-6xy+29y^2+6x-58y-151=0$
• B. $29x^2-6xy+21y^2+6x-58y-151=0$
• C. $21x^2-6xy+29y^2+58x-6y-151=0$
• D. $29x^2-6xy+21y^2+6x-58y+151=0$

Answer 1

The correct option is A $21x^2-6xy+29y^2+6x-58y-151=0$

Length of major axis $(x-3y+3=0)$ is $6$.
Length of minor axis $(3x+y-1=0)$ is $2\sqrt{6}$
Let variable point on ellipse be $P(x,y)$,
then equation of the required ellipse is
${\left(\frac{\text{distance of P from minor axis}}{\text{length of semi-major axis}}\right)}^2+{\left(\frac{\text{distance of P from major axis}}{\text{length of semi-minor axis}}\right)}^2=1$
${\left(\frac{\frac{3x+y-1}{\sqrt{9+1}}}{3}\right)}^2+{\left(\frac{\frac{x-3y+3}{\sqrt{1+9}}}{\sqrt{6}}\right)}^2=1$
$\Rightarrow \frac{(x-3y+3{)}^2}{60}+\frac{(3x+y-1{)}^2}{90}=1$
$\Rightarrow 3(x-3y+3{)}^2+2(3x+y-1{)}^2=180$
$\Rightarrow 21x^2-6xy+29y^2+6x-58y-151=0$