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This is equation of a curve


$A$ is constant

$T$ is a tangent of the curve from any point on it

$B$ is y-intercept of $T$

$C$ is x-intercept of $T$

prove that $B+C=$ Constant

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up vote 2 down vote accepted

The tangent at the point $(x_0,y_0)$ of the curve $f(x,y)=0$ has equation $$ (x-x_0)\frac{\partial f}{\partial x}(x_0,y_0)+(y-y_0)\frac{\partial f}{\partial x}=0 $$ and in this case we get $$ \frac{x-x_0}{2\sqrt{x_0\mathstrut}}+\frac{y-y_0}{2\sqrt{y_0\mathstrut}}=0 $$ Thus we can rewrite it as $$ \frac{x}{\sqrt{x_0\mathstrut}}+\frac{y}{\sqrt{y_0\mathstrut}}=\sqrt{A} $$ or $$ \frac{x}{\sqrt{A}\,\sqrt{x_0\mathstrut}}+ \frac{y}{\sqrt{A}\,\sqrt{y_0\mathstrut}}=1 $$ So the sum of the intercepts is $$ \sqrt{A\mathstrut}\,\sqrt{x_0\mathstrut}+ \sqrt{A\mathstrut}\,\sqrt{y_0\mathstrut}=A $$


Assume a point $(h,k)$ on the curve.

Differentiate given equation to get

$y^{'} = -\sqrt{\frac yx}$

Therefore equation of tangent using slope point form is

$y-k = m(x-h)$ , where m is $y^{'}$

Put $x=0 , y=0$ for C and B , and then add the two

It is also interesting to go in the opposite direction, i.e. to find the envelope of the segments for which $B+C=1$. Let $S_t$ be the segment with endpoints in $(0,t)$ and $(1-t,0)$, for any $t$ in the interval $(0,1)$. For any $\varepsilon>0$, $S_t$ and $S_{t+\varepsilon}$ meet in: $$ S_t\cap S_{t+\varepsilon} = \left((1-t)(1-t-\varepsilon),t\,(\varepsilon+t)\right) \tag{1} $$ hence by letting $\varepsilon\to 0$ we get that the parametric equation of the envelope is given by: $$ \left((1-t)^2,t^2\right) \tag{2}$$ that is a parabola with focus in $\left(\frac{1}{2},\frac{1}{2}\right)$ and directrix given by the line $x+y=0$.

In general, if we have a projective map $\varphi$ between two lines $l_1,l_2$, the envelope of the lines joining $P\in l_1$ with $\varphi(P)\in l_2$ is always a conic (I cannot recall the name of this theorem at the moment, but I am sure it is well-known).

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