Question

The curve on which the portion of its tangent is bisected at the curve point of intercation

Answer 1

To find the curve for which the part of tangent cut-off by the axes (the portion of the tangent between the coordinate axes) is bisected at the point of tangency.

Let xa+yb=1 be the tangent. It cuts the axes at (a,0) and (0,b). So the mid point of the part of tangent cut-off by the axes is (a2,b2). The slope of this tangent is −ba. (Since y=−bax+b).
Let the slope of the required curve at point (x,y)
given by dydx=f(x,y).

So we can say that f(a2,b2)=−ba⇒f(a2,b2)=−b/2a/2⇒f(x,y)=−yx.

Now f(x,y)=dydx=−yx⇒dyy=−dxx⇒log(y)=−log(x)+log(c)⇒xy=c

hope it helps you dear..