Question Number 37591 by mondodotto@gmail.com last updated on 15/Jun/18
$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{circle}}\:\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{y}}−\mathrm{8}=\mathrm{0} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} −\mathrm{24}\boldsymbol{{x}}+\boldsymbol{{hy}}=\mathrm{0}\:\boldsymbol{\mathrm{cut}}\:\boldsymbol{\mathrm{orthogonally}}, \\ $$$$\boldsymbol{\mathrm{determine}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{{h}}. \\ $$$$ \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 15/Jun/18
$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{y}−\mathrm{8}=\mathrm{0} \\ $$$${comparing}\:{with}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{2}{gx}+\mathrm{2}{fy}+{c}=\mathrm{0} \\ $$$${centre}\left(−{g},−{f}\right)\:\:{radius}=\sqrt{{g}^{\mathrm{2}} +{f}^{\mathrm{2}} −{c}} \\ $$$${o}_{\mathrm{1}} \left(\mathrm{0},\mathrm{1}\right)\:\:{r}_{\mathrm{1}} =\sqrt{\mathrm{0}^{\mathrm{2}} +\left(−\mathrm{1}\right)^{\mathrm{2}} −\left(−\mathrm{8}\right)}\:=\sqrt{\mathrm{9}}\:=\mathrm{3} \\ $$$${o}_{\mathrm{2}} \left(\mathrm{12},−\frac{{h}}{\mathrm{2}}\right)\:\:{r}_{\mathrm{2}} =\sqrt{\left(−\mathrm{12}\right)^{\mathrm{2}} +\left(\frac{{h}}{\mathrm{2}}\right)^{\mathrm{2}} } \\ $$$${r}_{\mathrm{2}} =\sqrt{\mathrm{144}+\frac{{h}^{\mathrm{2}} }{\mathrm{4}}}\: \\ $$$${o}_{\mathrm{1}} {o}_{\mathrm{2}} =\sqrt{\left(\mathrm{12}−\mathrm{0}\right)^{\mathrm{2}} +\left(−\frac{{h}}{\mathrm{2}}−\mathrm{1}\right)^{\mathrm{2}} }\:\: \\ $$$$=\sqrt{\mathrm{144}+\frac{{h}^{\mathrm{2}} }{\mathrm{4}}+{h}+\mathrm{1}}\:\:=\sqrt{\frac{{h}^{\mathrm{2}} }{\mathrm{4}}+{h}+\mathrm{145}}\: \\ $$$${r}_{\mathrm{1}} ^{\mathrm{2}} +{r}_{\mathrm{2}} ^{\mathrm{2}} =\left({o}_{\mathrm{1}} {o}_{\mathrm{2}} \right)^{\mathrm{2}} \\ $$$$\mathrm{9}+\mathrm{144}+\frac{{h}^{\mathrm{2}} }{\mathrm{4}}=\frac{{h}^{\mathrm{2}} }{\mathrm{4}}+{h}+\mathrm{145} \\ $$$${h}=\mathrm{8} \\ $$$${or}\:{mdthod}\: \\ $$$${clndition}\:{for}\:{orthogonslity} \\ $$$$\mathrm{2}{g}_{\mathrm{1}} {g}_{\mathrm{2}} +\mathrm{2}{f}_{\mathrm{1}} {f}_{\mathrm{2}} ={c}_{\mathrm{1}} +{c}_{\mathrm{2}} \\ $$$$\mathrm{2}×\mathrm{0}×\left(−\mathrm{12}\right)+\mathrm{2}×\left(−\mathrm{1}\right)×\frac{{h}}{\mathrm{2}}=−\mathrm{8}+\mathrm{0} \\ $$$$−{h}=−\mathrm{8} \\ $$$${h}=\mathrm{8} \\ $$$$ \\ $$