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Question Number 66256 by Mikael last updated on 11/Aug/19

$$Findallpoints(a,b)of{\mathbb{R}}^{2}suchthat$$$$through(a,b)passtwotangentlines$$$$tothegraphoff\left(x\right)=x^2.$$

Commented by kaivan.ahmadi last updated on 11/Aug/19

$$f^{\prime }\left(x\right)=2x$$$$let(x_0,y_0)beony=x^{2}andtangentline$$$$m=2x_0=\frac{b-y_0}{a-x_0}=\frac{b-x_0^2}{a-x_0}\Rightarrow$$$$2{ax}_0-2x_0^2=b-x_0^2\Rightarrow x_0^2-2{ax}_0+b=0\Rightarrow$$$$\mathrm{\Delta }mustbepositive$$$$4a^2-4b>0\Rightarrow a^2>b$$$$sotheanswersis(a,b)with{a}^2>b$$$$$$

Commented by mr W last updated on 12/Aug/19

$$fromeverypointoutsidetheparabola$$$$therecanbedrawntwotangentlines$$$$totheparabola,i.e.frompoint(x,y)$$$$withy<x^2,orb<a^2.$$

Commented by mr W last updated on 12/Aug/19

Commented by Mikael last updated on 12/Aug/19

$$thankyou.$$

Commented by Mikael last updated on 12/Aug/19

$$thankyouSir.$$

Commented by kaivan.ahmadi last updated on 13/Aug/19

$$howcanwedrowtwotangentfrom(1,4)?$$

Commented by mr W last updated on 14/Aug/19

$$no!point(1,4)isinsidetheparabola.$$$$4>1^2!$$

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