A tangent and a normal to a curve at any point P meet the x and y axes at A, B and C, D respectively. Find the equation of the curve passing through `(1, 0)` if the centre of circle through `O.C, P and B` lies on the line y = x (where O is the origin).

Options

A. `(dy)/(dx)=(y-x)/(y+x)`
B. `(dy)/(dx)=(y^(2)-x^(2))/(y^(2)+x^(2))`
C. `(dy)/(dx)=(x-y)/(xy)`
D. `(dy)/(dx)+(xy)/(x+y)`

Answer

a

Explanation

Let P be (x,y) .C is `(x+u (dy)/(dx),0)` and B is
`(0, y - x (dy)/(dx))` . Centre of the circle through O,C,P and B
has its centre at the point of BC. Let it be `(alpha,beta)` , then ` 2 alpha = x+y (dy)/(dx)` ltbr gt ` 2 beta = y - x (dy)/(dx)`
Now, ` (alpha,beta) ` lies on y = x

So , `y - x (dy)/(dx) = x+y rArr (dy)/(dx) rArr dy)/(dx) = (y-x)/(x+y)`