Three curves are given by C_1colon y=f(x), C_2colon

Three curves are given by $C_{1}\colon y=f(x)$, $C_{2}\colon y=\varphi(x)$, $C_{3}\colon y=\dfrac{f(x)+\lambda\varphi(x)}{1+\lambda}$, where $\lambda$ is a real parameter and $f,\varphi$ are differentiable. Show that the tangents to these three curves drawn at points with the same abscissa $x=a$ meet at a single point.

Step 1: The three tangents at $x=a$

Write $f_{0}=f(a)$, $\varphi_{0}=\varphi(a)$, $f_{0}'=f'(a)$, $\varphi_{0}'=\varphi'(a)$. Then

\[ \ell_{1}\colon\ y = f_{0}'x + (f_{0}-af_{0}'), \qquad \ell_{2}\colon\ y = \varphi_{0}'x + (\varphi_{0}-a\varphi_{0}'). \tag{1,2} \]

The value and derivative of $y_{3}=\dfrac{f+\lambda\varphi}{1+\lambda}$ at $x=a$ are the corresponding weighted averages, so

\[ \ell_{3}\colon\ y = \frac{f_{0}'+\lambda\varphi_{0}'}{1+\lambda}\,x + \frac{(f_{0}-af_{0}')+\lambda(\varphi_{0}-a\varphi_{0}')}{1+\lambda}. \tag{3} \]

Step 2: The key observation: $\ell_{3}$ is a weighted average of $\ell_{1}$ and $\ell_{2}$

Comparing (1)–(3) term by term, the slope of $\ell_{3}$ is exactly $\dfrac{k_{1}+\lambda k_{2}}{1+\lambda}$ and its intercept is exactly the same weighted average of the intercepts of $\ell_{1}$ and $\ell_{2}$. Hence, as functions of $x$,

\[ \ell_{3}(x) = \frac{\ell_{1}(x)+\lambda\,\ell_{2}(x)}{1+\lambda} \qquad \text{for every } x. \tag{4} \]

Step 3: The point common to $\ell_{1}$ and $\ell_{2}$ lies on $\ell_{3}$

Let $S=(x^{*},y^{*})$ be the intersection of $\ell_{1}$ and $\ell_{2}$ (it exists and is unique whenever $f_{0}'\neq\varphi_{0}'$, i.e.\ whenever the two tangents are not parallel), so $\ell_{1}(x^{*})=\ell_{2}(x^{*})=y^{*}$. Evaluating (4) at $x^{*}$:

\[ \ell_{3}(x^{*}) = \frac{\ell_{1}(x^{*})+\lambda\,\ell_{2}(x^{*})}{1+\lambda} = \frac{y^{*}+\lambda y^{*}}{1+\lambda} = y^{*}. \tag{5} \]

So $S$ lies on $\ell_{3}$ as well. $\blacksquare$

Step 4: Verification

\[f(x)=x^3-2x,\ \varphi(x)=\sin x,\ a=0.7,\ \lambda=2.3\text{: } \ell_3(x^*)-y^*=0 \text{ (numerically)}\ \checkmark\]

\[\text{Same functions, } a=-1.1,\ \lambda=0.4\text{: } \ell_3(x^*)-y^*=0\ \checkmark\]