Find the curve of a point (1, - 1) so that the tangent line at any point on the curve is bisected by the tangent line

Find the curve of a point (1, - 1) so that the tangent line at any point on the curve is bisected by the tangent line

Let the tangent point of the curve be (x, y), then according to the midpoint coordinate formula, it is easy to find the intersection points of the tangent line and X axis, Y axis are (2x, 0),
(0,2y), so the tangent slope is k = - Y / X. since the tangent slope of the curve is k = dy / DX, the differential equation can be obtained as
Dy / DX = - Y / X
In order to get XDY + YDX = 0, i.e. D (XY) = 0, integral, we get xy = C (C is any constant). Because the curve passes (1, - 1), we can get C = - 1 by substituting this point, so the equation of the curve is xy = - 1, i.e. y = - 1 / X