Find-all-points-a-b-of-R-2-such-that-through-a-b-pass-two-tangent-lines-to-the-graph-of-f-x-x-2-

Question Number 66256 by Mikael last updated on 11/Aug/19
$${Find}\:{all}\:{points}\:\left({a},\:{b}\right)\:{of}\:\mathbb{R}^{\mathrm{2}} \:{such}\:{that}\: \\$$$${through}\:\left({a},\:{b}\right)\:{pass}\:{two}\:{tangent}\:{lines} \\$$$${to}\:{the}\:{graph}\:{of}\:{f}\left({x}\right)={x}^{\mathrm{2}} . \\$$
Commented by kaivan.ahmadi last updated on 11/Aug/19
$${f}'\left({x}\right)=\mathrm{2}{x} \\$$$${let}\:\left({x}_{\mathrm{0}} ,{y}_{\mathrm{0}} \right)\:{be}\:{on}\:{y}={x}^{\mathrm{2}} \:{and}\:{tangent}\:{line} \\$$$${m}=\mathrm{2}{x}_{\mathrm{0}} =\frac{{b}−{y}_{\mathrm{0}} }{{a}−{x}_{\mathrm{0}} }=\frac{{b}−{x}_{\mathrm{0}} ^{\mathrm{2}} }{{a}−{x}_{\mathrm{0}} }\Rightarrow \\$$$$\mathrm{2}{ax}_{\mathrm{0}} −\mathrm{2}{x}_{\mathrm{0}} ^{\mathrm{2}} ={b}−{x}_{\mathrm{0}} ^{\mathrm{2}} \Rightarrow{x}_{\mathrm{0}} ^{\mathrm{2}} −\mathrm{2}{ax}_{\mathrm{0}} +{b}=\mathrm{0}\Rightarrow \\$$$$\Delta\:{must}\:{be}\:{positive} \\$$$$\mathrm{4}{a}^{\mathrm{2}} −\mathrm{4}{b}>\mathrm{0}\Rightarrow{a}^{\mathrm{2}} >{b} \\$$$${so}\:{the}\:{answers}\:{is}\:\left({a},{b}\right)\:{with}\:{a}^{\mathrm{2}} >{b} \\$$$$\\$$
Commented by mr W last updated on 12/Aug/19
$${from}\:{every}\:{point}\:{outside}\:{the}\:{parabola} \\$$$${there}\:{can}\:{be}\:{drawn}\:{two}\:{tangent}\:{lines} \\$$$${to}\:{the}\:{parabola},\:{i}.{e}.\:{from}\:{point}\:\left({x},{y}\right) \\$$$${with}\:{y}<{x}^{\mathrm{2}} ,\:{or}\:{b}<{a}^{\mathrm{2}} . \\$$
Commented by mr W last updated on 12/Aug/19
Commented by Mikael last updated on 12/Aug/19
$${thank}\:{you}. \\$$
Commented by Mikael last updated on 12/Aug/19
$${thank}\:{you}\:{Sir}. \\$$
Commented by kaivan.ahmadi last updated on 13/Aug/19
$${how}\:{can}\:{we}\:{drow}\:{two}\:{tangent}\:{from}\:\left(\mathrm{1},\mathrm{4}\right)? \\$$
Commented by mr W last updated on 14/Aug/19
$${no}!\:{point}\:\left(\mathrm{1},\mathrm{4}\right)\:{is}\:{inside}\:{the}\:{parabola}. \\$$$$\mathrm{4}>\mathrm{1}^{\mathrm{2}} \:\:! \\$$