Question Number 176147 by peter frank last updated on 13/Sep/22
Answered by mr W last updated on 13/Sep/22
Commented by mr W last updated on 13/Sep/22
$${the}\:{track}\:{of}\:{the}\:{ball}\:{is}\:{a}\:{parabola} \\ $$$${through}\:{points}\:{O},{A},{B},{C}.\:{it}'{s}\:{eqn}.\:{is} \\ $$$${y}={kx}\left({x}−{R}\right) \\ $$$${we}\:{have} \\ $$$${a}={kb}\left({b}−{R}\right)\:\:\:...\left({i}\right) \\ $$$${b}={ka}\left({a}−{R}\right)\:\:\:...\left({ii}\right) \\ $$$$\left({i}\right)/\left({ii}\right): \\ $$$$\frac{{a}}{{b}}=\frac{{b}\left({b}−{R}\right)}{{a}\left({a}−{R}\right)} \\ $$$${a}^{\mathrm{3}} −{a}^{\mathrm{2}} {R}={b}^{\mathrm{3}} −{b}^{\mathrm{2}} {R} \\ $$$$\left({b}^{\mathrm{2}} −{a}^{\mathrm{2}} \right){R}={b}^{\mathrm{3}} −{a}^{\mathrm{3}} =\left({b}−{a}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{ab}\right) \\ $$$$\Rightarrow{R}=\frac{{a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} }{{a}+{b}}\:\checkmark \\ $$
Commented by peter frank last updated on 14/Sep/22
$$\mathrm{thank}\:\mathrm{you} \\ $$
Commented by Tawa11 last updated on 15/Sep/22
$$\mathrm{Great}\:\mathrm{sir} \\ $$