Question

Find the curve whose gradient at any point P(x, y) on it is `(x - "a")/(y - "b")` and which passes through the origin

Answer 1

The gradient of the curve at P(x, y)

`("d"y)/("d"x) = (x - "a")/(y - "b")`

(y – b) dy = (x – a) dx

Integrating on both sides

`int (y - "b")  "d"y = int (x - "a")  "d"x`

⇒ `(y -- "b")^2/2 = (x - )^2/2 + "c"`

Multiply each term by 2

∴ (y – b)² = (x – a)² + 2c  .........(1)

Since the curve passes through the origin (0, 0)

Equation (1)

(0 – b)² = (0 – a)² + 2c

b² = a² + 2c

b² – a² = 2c  .......(2)

Substitute equation (2) and  (1)

(y – b)² = (x – a)² + b² – a²