What is the nature of the curve $x^{2} + y^{2} - 25 = 0$

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Solution

The equation of the given curve is
$F (x, y) = x^{2} + y^{2} - 25 = 0$
Here $F (x, - y) = x^{2} + {(- y)}^{2} - 25 = x^{2} + y^{2} - 25 = 0 = F (x, y)$
$F (- x, y) = {(- x)}^{2} + y^{2} - 25 = x^{2} + y^{2} - 25 = 0 = F (x, y)$
$F (- x, - y) = {(- x)}^{2} + {(- y)}^{2} - 25 = x^{2} + y^{2} - 25 = 0 = F (x, y)$
$∴$ the given curve is symmetrical about the $x -$ axis, $y -$ axis and about the origin.

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