Problem
GEO-B2-M09-P015 A Locus by Sum of Squares
#15
★★★☆☆ Level 3 of 5
Let \(A(-1,0)\), \(B(1,0)\). Find the locus of points \(P(x,y)\) for which \(PA^2+PB^2=10\).
Write \(PA^2\) and \(PB^2\), then combine like terms.
\(PA^2=(x+1)^2+y^2\), \(PB^2=(x-1)^2+y^2\). Their sum is \(2x^2+2y^2+2\). The condition \(PA^2+PB^2=10\) gives \(2x^2+2y^2+2=10\), that is \(x^2+y^2=4\). The locus is the circle with centre at the origin and radius \(2\).
The problem trains translating a geometric condition into an equation.