# The equation of the curve whose tangent makes a constant area with the axes

My turn: Let the equation of the curve is $$y=f(x)$$ Then the slope of the tangent is $$\frac{dy}{dx}$$ The equation of the tangent at some point $$x_o$$ Is $$y= f'(x_o) x + b$$ , $$b = f(x_o) -x_o f'(x_o)$$ Now the area of the triangle is $$A = 0.5 \frac{x_o f'(x_o) -f(x_o)}{f'(x_o)}\times (f(x_o) -x_o f'(x_o))$$ Now i think i need to put A as a constant an integrate the last equation with respect to $$x_o$$ to get the equation of th as a e curve but i am confused with how can i consider $$x_o$$ as a variable even though it is constant ?

• It is a difficult problem see my answer below. – Dr Zafar Ahmed DSc Oct 25 '19 at 19:37

I would write $$A=\frac{1}{2}\frac{(y_0-f'(x_0)x_0)^2}{|f'(x_0)|}$$ It is $$A=\frac{1}{2}\frac{(f(x_0)-x0_0f'(x_0))^2}{|f'(x_0)|}$$

• Okay, can i integrate this to get the equation of the curve ?Dr.Sonnhard Graubner – Hussien Mohamed Oct 25 '19 at 12:40
• I think $x_0-f'(x_0)x_0$ should be $f(x_0)-f'(x_0)x_0$. – David K Oct 25 '19 at 12:44
• Yes you have right, just corrected! – Dr. Sonnhard Graubner Oct 25 '19 at 12:52

It is difficult problem

The intercets on $$x$$ and $$y$$ axes by the tangent at the point $$(x,y)$$ are $$\Delta x=x-ydx/dy$$ and $$\Delta y=y-xdy/dx.$$ Then the area of the triangle with thes intercepts is $$|\frac{1}{2} (x-ydx/dy)(y-xdy/dx)|=A^2~~~(1)$$ Let us denote $$\frac{dy}{dx}=p$$ and let $$B^2p>0$$ Then we have $$x^2p^2+y^2-2xyp=B^2 p \implies |y-xp|= B\sqrt{p}~~~(2)$$ $$y=xp+B\sqrt{p} \implies y=x\frac{dy}{dx}+B\sqrt{\frac{dy}{dx}}~~~(3)$$ D.w.r.t.x to get $$y'=xy''+y'+ \frac{B}{2}\sqrt{\frac{dx}{dy}} y'' \implies y'' \left [x+\frac{B}{2}\sqrt{\frac{dx}{dy}} \right]=0~~~~(4)$$ $$y''=0$$ gives trivial solution $$(y'=C$$) which can be put in (3) to get the general solution of the first order ODE (3) which will be family of lines. For a fixed non-trivial solution of (3), we take $$\frac{B}{2}\sqrt{\frac{dx}{dy}}=-x \implies dy= \frac{B^2}{4x^2}dx \implies \int dy=\frac{B^2}{4} \int \frac{dx}{x^2}~~~~(5)$$ Finally, we get $$xy=-\frac{B^2}{4}~~~~(6),$$ the rectangular hyperbola. Note that this (6) is the singular solution of the Clairaut equaion (3). Here $$B$$ or $$A$$ is a fixed given constant.

Notte: it is rather simple to show the converse that the itercepts of the tangents to the curve $$xy=D^2$$ make a triangle of constant area.

• That is the same result i got but with different notations , would integrate it to get y ? Dr Zafar Ahmed DSc – Hussien Mohamed Oct 25 '19 at 13:36
• @Hussain Mohamed It is a difficult prob\em see my edited answer. – Dr Zafar Ahmed DSc Oct 25 '19 at 19:36