Question

Find the equation of a curve passing through the point (1, 1). If the tangent drawn at any point P(x, y) on the curve meets the co-ordinate axes at A and B such that P is the mid-point of AB.

Answer 1

Let P (x, y) be any point on the curve and AB be the tangent to the given curve at P.

P is the midpoint of AB  .....(Given)

∴ Coordinates of A and B are (2x, 0) and (0, 2y) respectively.

∴ Slope of the tangent AB = `(2y - 0)/(0 - 2x) = - y/x`

∴ `("d"y)/("d"x) = - y/x`

⇒ `("d"y)/y = -("d"x)/x`

Integrating both sides, we get

`int ("d"y)/y = -int ("d"x)/x`

⇒ log y = – log x + log c

⇒ log y + log x = log c

⇒ log yx = log c

∴ yx = c

Since, the curve passes through (1, 1)

∴ 1 × 1 = c

∴ c = 1

Hence, the required equation is xy = 1.