The equation of the curve in which the portion of the tangent included between the coordinate axes is bisected at the point of contact, is

Options

A. a parabola
B. an ellipse
C. a circle
D. a hyperbola

Answer

D

Explanation

The eqaution of the tangent at any point `P(x,y)` is
`Y-y=(dy)/(dx)(X-x)`
This cuts the coordinate axes at `A(x-y(dx)/(dy),0) and B(0,y-x(dy)/(dx))`.
It is given that `P(x,y)` is the mid-point of AB.
`therefore" "x-y(dx)/(dy)=2xand y -x(dy)/(dx)=2y`
`rArr" "x+y(dx)/(dy)=0 and y+x(dy)/(dx)=0`
`rArr" "x dy+ydx=0 and ydx+xdy =0`
`rArr" "d(xy)=0 rArr xy=C`
Clearly, it represents a rectangular hyperbola.