Question

The equation $(x-a{)}^2+(y-b{)}^2=k(px+qy+r{)}^2,$ where $(pa+qb+r)\ne 0$ represents

• A. a parabola for $k<\frac{1}{p^2+q^2}$
• B. an ellipse for $0<k<\frac{1}{p^2+q^2}$
• C. a hyperbola for $k>\frac{1}{p^2+q^2}$
• D. a point circle for $k=0$

Answer 1

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

a point circle for $k=0$

Here $(x-a{)}^2+(y-b{)}^2=k(px+qy+r{)}^2$ represents conic with $(a,b)$ as focus and $px+qy+r=0$ as directrix
$(x-a{)}^2+(y-b{)}^2=k(p^2+q^2){\left(\frac{px+qy+r}{\sqrt{p^2+q^2}}\right)}^2$
Here $e^2=k(p^2+q^2)$
For ellipse
$0<e^2<1\Rightarrow 0<k(p^2+q^2)<1\Rightarrow 0<k<\frac{1}{p^2+q^2}$
For parabola,
$e^2=1\Rightarrow k=\frac{1}{p^2+q^2}$
For hyperbola,
$e^2>1\Rightarrow k>\frac{1}{p^2+q^2}$
For $k=0,(x-a{)}^2+(y-b{)}^2=0\Rightarrow (x,y)=(a,b)$ a point circle