Question

The equation $3x^2+4y^2-18x+16y+43=k$

• A. represents empty set, if $k<0$
• B. represents an ellipse, if $k>0$
• C. represents a point, if $k=0$
• D. cannot represent a real pair of straight lines for any value of $k$

Answer 1

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

The correct options are
A represents empty set, if $k<0$
B represents an ellipse, if $k>0$
C represents a point, if $k=0$
D cannot represent a real pair of straight lines for any value of $k$

$3x^2+4y^2-18x+16y+43=k\Rightarrow 3(x-3{)}^2+4(y+2{)}^2=k$

If $k<0$ then no locus.

If $k>0$
$\frac{(x-3{)}^2}{k/3}+\frac{(y+2{)}^2}{k/4}=1$
Locus is an ellipse.

If $k=0$
$3(x-3{)}^2+4(y+2{)}^2=0\Rightarrow x=3,y=-2$
Locus is a point.