Problem
GEO-B2-M09-P002 A Circle with Diameter
#2
★☆☆☆☆ Level 1 of 5
Let \(A(-1,0)\), \(B(1,0)\), \(P(x,y)\), with \(P\ne A,B\). Prove that \(AP\perp BP\) if and only if \(x^2+y^2=1\).
Write the vectors \(\overrightarrow{PA}\) and \(\overrightarrow{PB}\).
\(\overrightarrow{PA}=(-1-x,-y)\), \(\overrightarrow{PB}=(1-x,-y)\). Their dot product is \((-1-x)(1-x)+y^2=x^2+y^2-1\). Perpendicularity is equivalent to this being zero, that is \(x^2+y^2=1\).
This is the coordinate version of the theorem about an angle subtending a diameter.