Question-196460

Question Number 196460 by peter frank last updated on 25/Aug/23

Commented by mr W last updated on 25/Aug/23

question is not clear. t_1 ,t_2 ,t_3 are three points. what do you mean with t_1 +t_2 +t_3 ?

$${question}\:{is}\:{not}\:{clear}.\: \\ $$$${t}_{\mathrm{1}} ,{t}_{\mathrm{2}} ,{t}_{\mathrm{3}} \:{are}\:{three}\:{points}.\:{what}\:{do}\:{you} \\ $$$${mean}\:{with}\:{t}_{\mathrm{1}} +{t}_{\mathrm{2}} +{t}_{\mathrm{3}} ? \\ $$

Commented by universe last updated on 25/Aug/23

i think t_(1 ) t_2 t_3 are slopes of normals

$${i}\:{think}\:{t}_{\mathrm{1}\:} {t}_{\mathrm{2}} \:{t}_{\mathrm{3}} \:{are}\:{slopes}\:{of}\:{normals} \\ $$

Commented by Spillover last updated on 26/Aug/23

Check you solution https://m.facebook.com/story.php?story_fbid=pfbid0hkqsvcktd24vmL7bxE8SHoaJdmVahXFrgpdvX3XSog1UD4cQCC5oNeqdbkLwQ1kKl&id=100084816875916&mibextid=Nif5oz

Commented by mr W last updated on 26/Aug/23

$${how}\:{could}\:{a}\:{question}\:{be}\:{answered}\: \\ $$$${when}\:{it}\:{is}\:{even}\:{not}\:{clear}\:{what}\:{the} \\ $$$${question}\:{is}? \\ $$

Commented by peter frank last updated on 26/Aug/23

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{spillover} \\ $$

Answered by mr W last updated on 27/Aug/23

Commented by mr W last updated on 27/Aug/23

say the equation of a normal from point P(h,k) is y=k+t(x−h) say the normal meets the parabola at point A(u,v). tan θ=(dx/dy)∣_(y=v) =(v/(2a))=−t ⇒v=−2at u=(v^2 /(4a))=(((−2at)^2 )/(4a))=at^2 v=k+t(u−h) −2at=k+t(at^2 −h) at^3 +(2a−h)t+k=0 three roots could exist for t, that means three normals through point (h,k) could exist^(∗)) . t_1 +t_2 +t_3 =(0/a)=0 ✓ ^(∗)) the condition that three normals exist is that the equation has three real roots, ⇒(−(k/(2a)))^2 +(((2a−h)/(3a)))^3 <0 ⇒((ak^2 )/4)+(((2a−h)^3 )/(27))<0 that means point P(h,k) must lie inside the green area: (example for a=0.5)

$${say}\:{the}\:{equation}\:{of}\:{a}\:{normal}\:{from}\: \\ $$$${point}\:{P}\left({h},{k}\right)\:{is}\: \\ $$$${y}={k}+{t}\left({x}−{h}\right) \\ $$$${say}\:{the}\:{normal}\:{meets}\:{the}\:{parabola} \\ $$$${at}\:{point}\:{A}\left({u},{v}\right). \\ $$$$\mathrm{tan}\:\theta=\frac{{dx}}{{dy}}\mid_{{y}={v}} =\frac{{v}}{\mathrm{2}{a}}=−{t} \\ $$$$\Rightarrow{v}=−\mathrm{2}{at} \\ $$$${u}=\frac{{v}^{\mathrm{2}} }{\mathrm{4}{a}}=\frac{\left(−\mathrm{2}{at}\right)^{\mathrm{2}} }{\mathrm{4}{a}}={at}^{\mathrm{2}} \\ $$$${v}={k}+{t}\left({u}−{h}\right) \\ $$$$−\mathrm{2}{at}={k}+{t}\left({at}^{\mathrm{2}} −{h}\right) \\ $$$${at}^{\mathrm{3}} +\left(\mathrm{2}{a}−{h}\right){t}+{k}=\mathrm{0} \\ $$$${three}\:{roots}\:{could}\:{exist}\:{for}\:{t},\:{that}\: \\ $$$${means}\:{three}\:{normals}\:{through}\: \\ $$$${point}\:\left({h},{k}\right)\:{could}\:{exist}\:^{\left.\ast\right)} . \\ $$$${t}_{\mathrm{1}} +{t}_{\mathrm{2}} +{t}_{\mathrm{3}} =\frac{\mathrm{0}}{{a}}=\mathrm{0}\:\checkmark \\ $$$$\:^{\left.\ast\right)} \:{the}\:{condition}\:{that}\:{three}\:{normals} \\ $$$${exist}\:{is}\:{that}\:{the}\:{equation}\:{has}\:{three}\: \\ $$$${real}\:{roots}, \\ $$$$\Rightarrow\left(−\frac{{k}}{\mathrm{2}{a}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{2}{a}−{h}}{\mathrm{3}{a}}\right)^{\mathrm{3}} <\mathrm{0} \\ $$$$\Rightarrow\frac{{ak}^{\mathrm{2}} }{\mathrm{4}}+\frac{\left(\mathrm{2}{a}−{h}\right)^{\mathrm{3}} }{\mathrm{27}}<\mathrm{0} \\ $$$${that}\:{means}\:{point}\:{P}\left({h},{k}\right)\:{must}\:{lie} \\ $$$${inside}\:{the}\:{green}\:{area}: \\ $$$$\left({example}\:{for}\:{a}=\mathrm{0}.\mathrm{5}\right) \\ $$

Commented by mr W last updated on 27/Aug/23