Question

The equation $2x^2+3y^2-8x-18y+35=k$, represents

• A. no locus if $k>0$
• B. an ellipse if $k>0$
• C. a point if $k=0$
• D. a hyperbola if $k>0$

Answer 1

Answer: (B), (C)

a point if $k=0$
an ellipse if $k>0$

$2x^2+3y^2-8x-18y+35=k$
$2(x^2-4x)+3(y^2-6y)+35=k$
$2(x^2-4x+4)+3(y^2-6y+9)=k$
$2(x-2{)}^2+3(y-3{)}^2=k$ ....(1)
Now if $k=0\left(1\right)$ represents point $(x=2,y=3)$
If $k>0$ clearly (1) is a matching with standard equation of ellipse
$\frac{(x-2{)}^2}{k/2}+\frac{(y-3{)}^2}{k/3}=1$