Question

If the equation $\sqrt{(x-1{)}^2+(y-2{)}^2}=\sqrt{\lambda }|x+y+3|$ represents a parabola, then

• A. $\lambda =1$
• B. $\lambda =\frac{1}{3}$
• C. Coordinates of focus are $(1,2)$
• D. Coordinates of focus are $(1,0)$

Answer 1

The correct option is C Coordinates of focus are $(1,2)$

The given equation is $\sqrt{(x-1{)}^2+(y-2{)}^2}=\sqrt{\lambda }|x+y+3|$
Squaring both sides, we get
$(x-1{)}^2+(y-2{)}^2=\lambda (x+y+3{)}^2$
$\Rightarrow (x-1{)}^2+(y-2{)}^2=2\lambda {\left(\frac{x+y+3}{\sqrt{2}}\right)}^2$
Above equation represents parabola if
$2\lambda =1$
$\Rightarrow \lambda =\frac{1}{2}$
The focus of the parabola is $(1,2)$