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At which point does the normal to the hyperbola xy = 4 at (2, 2) intersects the hyperbola again?

At which point does the normal to the hyperbola xy = 4 at (2, 2) intersects the hyperbola again? Correct Answer (-2, -2)

Equation of the given hyperbola is, xy = 4 ……….(1) Differentiating both side of (1) with respect to y, we get, y*(dx/dy) + x(1) = 0 Or dx/dy = -(x/y) Thus, the required equation of the normal to the hyperbola at (2, 2) is, y – 2 = -(2, 2) (x – 2) = -(-2/2)(x – 2) So, from here, y – 2 = x – 2 Or x – y = 0 ……….(2) Solving the equation (1) and (2) we get, x = 2 and y = 2 or x = -2 and y = -2 Thus, the line (2) intersects the hyperbola (1) at (2, 2) and (-2, -2). Hence, the evident is that the normal at (2, 2) to the hyperbola (1) again intersects it at (-2, -2).

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