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(A) Let $A(x_1, y_1)$,$B(x_2, y_2)$,$C(x_3, y_3)$ and $P(x, y)$ be points in the $xy$-plane.Given the condition $PA^2 + PB^2 = 2PC^2$,we substitute th

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Straight line

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(A) Let $A(x_1, y_1)$,$B(x_2, y_2)$,$C(x_3, y_3)$ and $P(x, y)$ be points in the $xy$-plane.Given the condition $PA^2 + PB^2 = 2PC^2$,we substitute the distance formula:$(x - x_1)^2 + (y - y_1)^2 + (x - x_2)^2 + (y - y_2)^2 = 2[(x - x_3)^2 + (y - y_3)^2]$Expanding the squares:$(x^2 - 2xx_1 + x_1^2 + y^2 - 2yy_1 + y_1^2) + (x^2 - 2xx_2 + x_2^2 + y^2 - 2yy_2 + y_2^2) = 2(x^2 - 2xx_3 + x_3^2 + y^2 - 2yy_3 + y_3^2)$$2x^2 + 2y^2 - 2x(x_1 + x_2) - 2y(y_1 + y_2) + (x_1^2 + x_2^2 + y_1^2 + y_2^2) = 2x^2 + 2y^2 - 4xx_3 - 4yy_3 + 2(x_3^2 + y_3^2)$The $2x^2$ and $2y^2$ terms cancel out on both sides:$-2x(x_1 + x_2) - 2y(y_1 + y_2) + (x_1^2 + x_2^2 + y_1^2 + y_2^2) = -4xx_3 - 4yy_3 + 2(x_3^2 + y_3^2)$Rearranging into the form $ax + by + c = 0$:$x(4x_3 - 2x_1 - 2x_2) + y(4y_3 - 2y_1 - 2y_2) + (x_1^2 + x_2^2 + y_1^2 + y_2^2 - 2x_3^2 - 2y_3^2) = 0$Since this is a linear equation in $x$ and $y$,the locus of $P$ is a straight line.

Let $A, B$ and $C$ be three points in a plane. The locus of a point $P$ moving such that $PA^2 + PB^2 = 2PC^2$ is a

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