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Question

The ends of hypotenuse of a right angled triangle are (a,0),(−a,0) then the locus of third vertex is:

A
x2y2=2a2
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B
x2y2=a2
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C
x2+y2=a2
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D
x2y2=4a2
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Solution

The correct option is B x2+y2=a2
Let the third vertex be P(x,y)

Let C=(a,0) and B=(a,0)
Since PCB form a right angled triangle right angled at P, then application of Pythagoras theorem, gives us
(PC)2+(PB)2=(BC)2
[(xa)2+y2]+[(x+a)2+y2]=4a2
[(xa)2+(x+a)2)+2y2=4a2
2x2+2y2+2a2=4a2
2x2+2y2=2a2
x2+y2=a2
This represents a circle.

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