A normal is drawn at a point `P(x , y)` of a curve. It meets the x-axis and the y-axis in point `A` AND `B ,` respectively, such that `1/(O A)+1/(O B)` =1, where `O` is the origin. Find the equation of such a curve passing through `(5. 4)`

Options

A. `x^(2)+y^(2)-4x-6y+8=0`
B. `x^(2)+y^(2)-2x-3y+3=0`
C. `2x^(2)+2y^(2)-x-y-2=0`
D. `x^(2)+y^(2)-4x+6y-4=0`

Answer

A, D

Explanation

Equation of normal at P
`y^(')-y=-(1)/(m)(x^(-1)-x)`
`A(x+y(dy)/(dx),0),B(0,y+(xdx)/(dy))`
`(2dx)/(xdx+ydy)+-(3dy)/(xdx+ydy)=1`
`2dx+3dy=xdx+ydy`
`2x+3y=(x^(2))/(2)+(y^(2))/(2)+C`
`C=4`
`x^(2)-y^(2)-4x-6y+8=0`