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Question Number 80795 by mr W last updated on 06/Feb/20

Commented by mr W last updated on 06/Feb/20

Find the equation of the parabola, i.e.  the value k as well as the corresponding  coordinates of points A,B,C such that  the areas I, II, III, IV  are equal.

$${Find}\:{the}\:{equation}\:{of}\:{the}\:{parabola},\:{i}.{e}. \\ $$$${the}\:{value}\:{k}\:{as}\:{well}\:{as}\:{the}\:{corresponding} \\ $$$${coordinates}\:{of}\:{points}\:{A},{B},{C}\:{such}\:{that} \\ $$$${the}\:{areas}\:{I},\:{II},\:{III},\:{IV}\:\:{are}\:{equal}. \\ $$

Commented by MJS last updated on 06/Feb/20

I think there′s no solution  you showed there′s no solution for k=1  y=x^2   let x_1 =(x/a)∧y_1 =(y/a) ⇔ x=ax_1 ∧y=ay_1   y_1 =ax_1 ^2   ⇒  all parabolas y_1 =ax_1 ^2  are similar to y=x^2

$$\mathrm{I}\:\mathrm{think}\:\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{solution} \\ $$$$\mathrm{you}\:\mathrm{showed}\:\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{solution}\:\mathrm{for}\:{k}=\mathrm{1} \\ $$$${y}={x}^{\mathrm{2}} \\ $$$$\mathrm{let}\:{x}_{\mathrm{1}} =\frac{{x}}{{a}}\wedge{y}_{\mathrm{1}} =\frac{{y}}{{a}}\:\Leftrightarrow\:{x}={ax}_{\mathrm{1}} \wedge{y}={ay}_{\mathrm{1}} \\ $$$${y}_{\mathrm{1}} ={ax}_{\mathrm{1}} ^{\mathrm{2}} \\ $$$$\Rightarrow \\ $$$$\mathrm{all}\:\mathrm{parabolas}\:{y}_{\mathrm{1}} ={ax}_{\mathrm{1}} ^{\mathrm{2}} \:\mathrm{are}\:\mathrm{similar}\:\mathrm{to}\:{y}={x}^{\mathrm{2}} \\ $$

Commented by mr W last updated on 07/Feb/20

you are right sir!

$${you}\:{are}\:{right}\:{sir}! \\ $$

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