Does the point (–2.5, 3.5) lie inside, outside or on the circle x² + y² = 25?

Let us write it as x² + y² – 25 = 0 and take f(x, y) = x² + y² – 25. For the points lying on the circle, f(x, y) = 0, for the points inside circle, it has sign as the sign obtained by putting the value of the coordinates of the centre in the expression. If it has opposite sign then it lies outside circle.

F(0, 0) = 0 + 0 – 25 = -ve.

Given point is (-2.5, 3.5)

Hence f(-2.5, 3.5) = (-2.5)² + (3.5)² – 25

                         = 6.25 + 12.5 – 25

                         = 18.5 – 25

                         = -6.5 = -ve

Hence, the given point lies inside the circle.