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Question

The equation (xa)2+(yb)2=k(px+qy+r)2, where (pa+qb+r)0 represents

A
a parabola for k<1p2+q2
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B
an ellipse for 0<k<1p2+q2
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C
a hyperbola for k>1p2+q2
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D
a point circle for k=0
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Solution

The correct option is D a point circle for k=0
Here (xa)2+(yb)2=k(px+qy+r)2 represents conic with (a,b) as focus and px+qy+r=0 as directrix
(xa)2+(yb)2=k(p2+q2)(px+qy+rp2+q2)2
Here e2=k(p2+q2)
For ellipse
0<e2<10<k(p2+q2)<10<k<1p2+q2
For parabola,
e2=1k=1p2+q2
For hyperbola,
e2>1k>1p2+q2
For k=0,(xa)2+(yb)2=0(x,y)=(a,b) a point circle

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