Find the curves for which the midpoint of the segment of the…
Find the curves for which the midpoint of the segment of the tangent line between the coordinate axes always lies on the parabola \(y^2=2px\).
Explanation of Method
Express the midpoint \((\xi,\eta)\) of the tangent's intercepts in terms of \(x,y,y'\), impose \(\eta^2=2p\xi\), and reduce algebraically to a Clairaut equation, solved in the usual way (differentiate, split into the two factors). The reduction to the Clairaut equation is verified with Sympy by clearing denominators and polynomial division. The source's closing claim — that the resulting line family is "the family of tangents to \(y^2=2px\)" — is checked independently by computing the actual envelope of the solution family via the standard \(F=0,\ \partial F/\partial c=0\) elimination (Sympy \(\texttt{resultant}\)), and is found to be false: the true envelope is a different parabola.
1. Midpoint condition reduces exactly to the Clairaut equation
With intercepts \(A(x-\tfrac{y}{w},0)\), \(B(0,y-xw)\) (\(w=y'\)), the midpoint is \(\xi=\tfrac12(x-\tfrac{y}{w})\), \(\eta=\tfrac12(y-xw)\). Substituting into \(\eta^2-2p\xi=0\), clearing denominators (multiply by \(4w\)), and factoring with Sympy's polynomial division: writing \(u=y-xw\), the cleared numerator factors exactly as \(u\cdot(wu+4p)\) — confirmed by Sympy's \(\texttt{div}\) returning quotient \(u\) and remainder \(0\) when dividing by \((wu+4p)\). Discarding the degenerate factor \(u=0\) (tangent through the origin) and keeping \(wu+4p=0\), i.e. \(w(y-xw)=-4p\), rearranges to \[ y = xw - \frac{4p}{w}, \] exactly reproducing the source's Clairaut equation with no algebra error.
2. Solve the Clairaut equation
Differentiating \(y=xw-4p/w\) with respect to \(x\) gives \(w'(x+4p/w^2)=0\), so either \(w'=0\) (general solution \(y=cx-4p/c\), a one-parameter family of lines) or \(x=-4p/w^2\) (singular solution). Both branches match the source.
3. Check the source's closing claim: is \(y=cx-4p/c\) really tangent to \(y^2=2px\)?
The standard tangent-line family of \(y^2=2px\) is \(y=mx+\tfrac{p}{2m}\); Sympy confirms this by eliminating \(m\) between \(F=mx+\tfrac{p}{2m}-y=0\) and \(\partial F/\partial m=0\) via \(\texttt{resultant}\), which returns \(8px(2px-y^2)\) — the nontrivial factor \(2px-y^2=0\) is indeed the original parabola, as expected.
Running the identical elimination on the source's family \(y=cx-4p/c\) (\(F=cx-4p/c-y\), eliminate \(c\) between \(F=0\) and \(\partial F/\partial c=x+4p/c^2=0\)) gives, via Sympy's \(\texttt{resultant}\), \[ 4px(16px+y^2)=0, \] whose nontrivial factor is \(y^2=-16px\) — not \(y^2=2px\). A second, independent computation confirms this: substituting the singular-solution condition \(w^2=-4p/x\) directly into \(y=xw-4p/w\) and simplifying \(y^2\) using \(w^2=-4p/x\) (Sympy) gives \(y^2=-16px\) exactly, matching the envelope computation. So the source's closing "note" is false: the lines \(y=cx-4p/c\) are tangent to the parabola \(y^2=-16px\) — a different parabola from the given \(y^2=2px\) (same axis, opposite orientation, and a different latus rectum, \(16p\) vs. \(2p\)) — not to \(y^2=2px\) itself. (The two parabolas coincide only in the trivial case \(p=0\).) This is a real, if secondary, coefficient-mismatch error in the source, separate from the main derivation, which remains correct through the Clairaut equation and both of its solution branches.
Result
Final answer:
The derivation of the Clairaut equation \(y=xw-\tfrac{4p}{w}\) and both of its solution branches are confirmed correct. The source's closing identification of the envelope, however, is wrong and is corrected here: \[ \textbf{General solution (lines):}\quad \boxed{\ y = cx - \frac{4p}{c}\ } \] \[ \textbf{Singular solution (envelope), computed and verified two independent ways:}\quad \boxed{\ y^2 = -16px\ } \] i.e. the tangent lines whose midpoint lies on \(y^2=2px\) form the family \(y=cx-4p/c\), and this family is itself tangent to the different parabola \(y^2=-16px\) — not, as the source incorrectly asserts, to the original parabola \(y^2=2px\).