Home › ACSEE (Form 6) › Advanced Mathematics › Hyperbolic Functions › If $m\sinh x+n\cosh x=h$ has equal roots, $h$ ex…
If $m\sinh x+n\cosh x=h$ has equal roots, $h$ expressed in terms of $m$ and $n$ is:
A$h=\pm\sqrt{n^2-m^2}$
B$h=\pm\sqrt{m^2-n^2}$
C$h=\pm\sqrt{m^2+n^2}$
D$h=\pm\sqrt{n^2+m^2}$
Answer & Solution
Correct answer: A. $h=\pm\sqrt{n^2-m^2}$
1. Write $\cosh x=\tfrac12(e^{x}+e^{-x})$ and $\sinh x=\tfrac12(e^{x}-e^{-x})$.
2. Substituting into $m\sinh x+n\cosh x=h$ and clearing denominators gives $(m+n)e^{2x}-2he^{x}+(n-m)=0$.
3. This is a quadratic in $e^{x}$. Equal roots require the discriminant to vanish: $(-2h)^2-4(m+n)(n-m)=0$.
4. So $4h^2=4(n^2-m^2)$, giving $h^2=n^2-m^2$ and $h=\pm\sqrt{n^2-m^2}$.
5. The option with $m^2-n^2$ reverses the sign of the product $(m+n)(n-m)$; the two with a plus sign treat the constant term as $(n+m)$.
_Source: NECTA ACSEE 2023 Advanced Mathematics 142/1, Question 2: Hyperbolic Functions_