Question Number 74300 by ~blr237~ last updated on 21/Nov/19
![Let consider γ :I→R^2 a parametric curve 1)Prove that if a<b and γ(a)≠γ(b) then there exist t_0 ∈]a,b[ such as γ′(t_0 ) is colinear to γ(b)−γ(a) 2)Show that if γ is regular and the function f :I→R t→f(t)=∣∣γ(t)−O(0,0) ∣∣ is maximal in t_0 ∈I Then ∣K_γ (t_0 )∣≥(1/(f(t_0 )))](https://www.tinkutara.com/question/Q74300.png)
$${Let}\:{consider}\:\:\gamma\:\::{I}\rightarrow\mathbb{R}^{\mathrm{2}} \:\:{a}\:{parametric}\:{curve}\: \\ $$$$\left.\mathrm{1}\left.\right){Prove}\:{that}\:{if}\:\:{a}<{b}\:\:{and}\:\:\gamma\left({a}\right)\neq\gamma\left({b}\right)\:{then}\:{there}\:{exist}\:\:{t}_{\mathrm{0}} \in\right]{a},{b}\left[\:\:\right. \\ $$$${such}\:{as}\:\:\gamma'\left({t}_{\mathrm{0}} \right)\:\:{is}\:{colinear}\:{to}\:\gamma\left({b}\right)−\gamma\left({a}\right)\: \\ $$$$\left.\mathrm{2}\right){Show}\:{that}\:{if}\:\:\gamma\:{is}\:{regular}\:{and}\:{the}\:\:{function}\:{f}\::{I}\rightarrow\mathbb{R}\:\:\:\:{t}\rightarrow{f}\left({t}\right)=\mid\mid\gamma\left({t}\right)−{O}\left(\mathrm{0},\mathrm{0}\right)\:\mid\mid\:\:{is}\:{maximal}\:{in}\:{t}_{\mathrm{0}} \in{I} \\ $$$${Then}\:\:\mid{K}_{\gamma} \left({t}_{\mathrm{0}} \right)\mid\geqslant\frac{\mathrm{1}}{{f}\left({t}_{\mathrm{0}} \right)} \\ $$
Answered by mind is power last updated on 21/Nov/19
![γ(t)=(x(t),y(t)) γ(a)≠γ(b) ⇒x(a)≠x(b) or y(a)≠y(b) x(a)≠x(b) g(t)=y(t)(x(a)−x(b)−x(t)(y(a)−y(b)) g(a)=−y(a)x(b)+x(a)y(b) g(b)=y(b)x(a)−x(b)y(a)=g(a) g(a)=g(b) mean values ⇒∃t_0 ∈]a,b[∣g′(t_0 )=0 g′(t_0 )=y′(t_0 )(x(a)−x(b))−x′(t_0 ).(y(a)−y(b))=0 since γ(a)≠γ(b) we have 3 cases if x(a)=x(b)⇒x′(t_0 )=0 x(a)≠x(b) & y(a)−y(b)=0⇒y′(t_0 )=0 x(a)#x(b)&y(a)≠y(b)⇒ ⇔((y′(t_0 ))/(x′(t_0 )))=((y(a)−y(b))/(x(a)−x(b))) ((y′(t_0 ))/(x′(t_0 ))) is coeficuent of tangent in t_o ⇒in t_0 ,γ′(t_0 ) is colinear to γ(b)−γ(a) 2) calcule corbur](https://www.tinkutara.com/question/Q74306.png)
$$\gamma\left({t}\right)=\left({x}\left({t}\right),{y}\left({t}\right)\right) \\ $$$$\gamma\left({a}\right)\neq\gamma\left({b}\right) \\ $$$$\Rightarrow{x}\left({a}\right)\neq{x}\left({b}\right)\:\:{or}\:{y}\left({a}\right)\neq{y}\left({b}\right) \\ $$$${x}\left({a}\right)\neq{x}\left({b}\right) \\ $$$${g}\left({t}\right)={y}\left({t}\right)\left({x}\left({a}\right)−{x}\left({b}\right)−{x}\left({t}\right)\left({y}\left({a}\right)−{y}\left({b}\right)\right)\right. \\ $$$${g}\left({a}\right)=−{y}\left({a}\right){x}\left({b}\right)+{x}\left({a}\right){y}\left({b}\right) \\ $$$${g}\left({b}\right)={y}\left({b}\right){x}\left({a}\right)−{x}\left({b}\right){y}\left({a}\right)={g}\left({a}\right) \\ $$$$\left.{g}\left({a}\right)={g}\left({b}\right)\:{mean}\:{values}\:\Rightarrow\exists{t}_{\mathrm{0}} \in\right]{a},{b}\left[\mid{g}'\left({t}_{\mathrm{0}} \right)=\mathrm{0}\right. \\ $$$${g}'\left({t}_{\mathrm{0}} \right)={y}'\left({t}_{\mathrm{0}} \right)\left({x}\left({a}\right)−{x}\left({b}\right)\right)−{x}'\left({t}_{\mathrm{0}} \right).\left({y}\left({a}\right)−{y}\left({b}\right)\right)=\mathrm{0} \\ $$$${since}\:\gamma\left({a}\right)\neq\gamma\left({b}\right)\:{we}\:{have}\:\mathrm{3}\:{cases} \\ $$$${if}\:{x}\left({a}\right)={x}\left({b}\right)\Rightarrow{x}'\left({t}_{\mathrm{0}} \right)=\mathrm{0} \\ $$$${x}\left({a}\right)\neq{x}\left({b}\right)\:\&\:{y}\left({a}\right)−{y}\left({b}\right)=\mathrm{0}\Rightarrow{y}'\left({t}_{\mathrm{0}} \right)=\mathrm{0} \\ $$$${x}\left({a}\right)#{x}\left({b}\right)\&{y}\left({a}\right)\neq{y}\left({b}\right)\Rightarrow \\ $$$$\Leftrightarrow\frac{{y}'\left({t}_{\mathrm{0}} \right)}{{x}'\left({t}_{\mathrm{0}} \right)}=\frac{{y}\left({a}\right)−{y}\left({b}\right)}{{x}\left({a}\right)−{x}\left({b}\right)} \\ $$$$\frac{{y}'\left({t}_{\mathrm{0}} \right)}{{x}'\left({t}_{\mathrm{0}} \right)}\:\:{is}\:{coeficuent}\:{of}\:{tangent}\:{in}\:{t}_{{o}} \\ $$$$\Rightarrow{in}\:{t}_{\mathrm{0}} ,\gamma'\left({t}_{\mathrm{0}} \right)\:\:{is}\:{colinear}\:{to}\:\gamma\left({b}\right)−\gamma\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calcule}\:\:{corbur} \\ $$$$ \\ $$$$ \\ $$