Prove that the subnormal of the curve y²=2a ln x+c is
Prove that the subnormal of the curve $y^2=2a\ln x+c$ is inversely proportional to the abscissa.
Explanation of Method
Implicit differentiation eliminates y altogether, leaving the subnormal a pure function of x.
Step 1: The subnormal formula
For $y=f(x)$, the subnormal at $(x_0,y_0)$ is $S_N=y_0\,y_0'$.
Step 2: Implicit differentiation
Differentiating $y^2=2a\ln x+c$ with respect to $x$: \[2y\,y' = \frac{2a}{x} \;\Longrightarrow\; y\,y' = \frac{a}{x}. \tag{1}\]
Step 3: Verification
\[a=3,\ c=1,\ x_0=5\text{: } y_0=\sqrt{2a\ln x_0+c}\approx3.29876,\ y_0'=a/(x_0y_0)\approx0.18189,\ y_0y_0'\approx0.60000=a/x_0\ \checkmark\]
Result
Final answer:
\[\boxed{\,S_N = \frac{a}{x}\,}\]
The left side of (1) is exactly the subnormal, so
\[ S_N = y\,y' = \frac{a}{x}, \] and $S_N\cdot x=a=\text{const}$: the subnormal is inversely proportional to the abscissa, with constant of proportionality $a$. $\blacksquare$