Let C be the circle of radius unity centred at the origin.If two positive numbers $x_1$ and $x_2$ are such that the line passing through $(x_1,-1)$ and $(x_2+1)$ is tangent to $C$ then ?

The line passes through the points (x_1, -1) and (x_2, 1), so its slope m is (1 - (-1)) divided by (x_2 - x_1), which equals 2/(x_2 - x_1). Using the point-slope form from (x_2, 1), the equation of the line is y - 1 = m(x - x_2), which rearranges to mx - y + (1 - mx_2) = 0. Since this line is tangent to the unit circle x^2+y^2=1, the distance from the origin (0,0) to the line equals 1, giving the equation |1 - mx_2| = sqrt(m^2 + 1), which implies (1 - mx_2)^2 = m^2 + 1. Expanding this gives m^2x_2^2 - 2mx_2 = m^2, and substituting 2/(x_2 - x_1) for m results in x_1 times x_2 = 1.

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