Question Number 43783 by ajfour last updated on 15/Sep/18
Commented by ajfour last updated on 16/Sep/18
$${can}\:{an}\:{app}\:{plot}\:{it}\:{if}\:{it}\:{be}\:{unique}? \\ $$
Commented by ajfour last updated on 15/Sep/18
$${Should}\:{it}\:{be}\:{unique}? \\ $$
Commented by MrW3 last updated on 15/Sep/18
$${I}\:{don}'{t}\:{think}\:{it}'{s}\:{unique}. \\ $$
Commented by ajfour last updated on 15/Sep/18
$${i}\:{have}\:{a}\:{notion},\:{it}'{ll}\:{be}\:{a}\:{single} \\ $$$${equation}. \\ $$
Commented by MJS last updated on 16/Sep/18
$$\mathrm{parabola}\:\cap\:\mathrm{circle}\:\mathrm{has}\:\mathrm{got}\:\mathrm{4}\:\mathrm{solutions},\:\mathrm{real} \\ $$$$\mathrm{or}\:\mathrm{complex}\:\Rightarrow\:\mathrm{we}'\mathrm{ll}\:\mathrm{get}\:\mathrm{polynomes}\:\mathrm{of}\:\mathrm{4}^{\mathrm{th}} \:\mathrm{degree} \\ $$$$\mathrm{including}\:\mathrm{3}\:\mathrm{unknown}\:\mathrm{and}\:\mathrm{1}\:\mathrm{parameter} \\ $$$$ \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{6}\:\mathrm{unknown}\:{p},\:{q},\:{x}_{{A}} ,\:{y}_{{A}} ,\:{x}_{{B}} ,\:{y}_{{B}} \\ $$$$\mathrm{par}:\:{y}={q}−{px}^{\mathrm{2}} \\ $$$$\mathrm{par}\:\cap\:\mathrm{red}\:\mathrm{circle}\:={A}=\begin{pmatrix}{{x}_{{A}} }\\{{y}_{{A}} }\end{pmatrix} \\ $$$$\mathrm{par}\:\cap\:\mathrm{blue}\:\mathrm{circle}\:=\:{B}=\begin{pmatrix}{{x}_{{B}} }\\{{y}_{{B}} }\end{pmatrix} \\ $$$$\mathrm{but}\:\mathrm{I}\:\mathrm{can}'\mathrm{t}\:\mathrm{see}\:\mathrm{more}\:\mathrm{than}\:\mathrm{2},\:\mathrm{maybe}\:\mathrm{4}\:\mathrm{equations} \\ $$$$\mathrm{looks}\:\mathrm{bad}...\:\mathrm{but}\:\mathrm{I}'\mathrm{m}\:\mathrm{usually}\:\mathrm{too}\:\mathrm{pessimistic} \\ $$
Answered by ajfour last updated on 17/Sep/18
$${let}\:{parabola}\:{eq}.\:{be}\: \\ $$$$\:\:\:\:\:\:\boldsymbol{{y}}\:=\:\boldsymbol{{c}}−\boldsymbol{{Ax}}^{\mathrm{2}} \\ $$$${let}\:{point}\:{of}\:{tangency}\:{with}\:{circle} \\ $$$${with}\:{centre}\:\left(\boldsymbol{{r}},\boldsymbol{{r}}\right)\:{and}\:{radius}\:\boldsymbol{{r}}\:{be} \\ $$$$\:\:\:\:{P}\:\left(\boldsymbol{{h}},\:\boldsymbol{{c}}−\boldsymbol{{Ah}}^{\mathrm{2}} \right) \\ $$$${slope}\:{of}\:{tangent}\:{to}\:{parabola}\:= \\ $$$${slope}\:{of}\:{tangent}\:{to}\:{circle}\: \\ $$$$\Rightarrow\:\:\:−\mathrm{2}\boldsymbol{{Ah}}\:=\:\frac{\boldsymbol{{c}}−\boldsymbol{{Ah}}^{\mathrm{2}} −\boldsymbol{{r}}}{\boldsymbol{{h}}−\boldsymbol{{r}}} \\ $$$${Also} \\ $$$$\:\:\left({h}−{r}\right)^{\mathrm{2}} +\left({c}−{Ah}^{\mathrm{2}} −{r}\right)^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$$\Rightarrow\:\:\left({h}−{r}\right)^{\mathrm{2}} +\frac{\left({h}−{r}\right)^{\mathrm{2}} }{\mathrm{4}{A}^{\mathrm{2}} {h}^{\mathrm{2}} }\:=\:{r}^{\mathrm{2}} \\ $$$$\Rightarrow\:\left({h}−{r}\right)^{\mathrm{2}} \:=\:\mathrm{4}{A}^{\mathrm{2}} {h}^{\mathrm{3}} \left(\mathrm{2}{r}−{h}\right) \\ $$$$...... \\ $$