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Question Number 136284 by liberty last updated on 20/Mar/21

What is shortest distance   from A(3,8) to the circle   x^2 +y^2 +4x−6y = 12

$${What}\:{is}\:{shortest}\:{distance}\: \\ $$$${from}\:{A}\left(\mathrm{3},\mathrm{8}\right)\:{to}\:{the}\:{circle}\: \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{6}{y}\:=\:\mathrm{12}\: \\ $$

Answered by EDWIN88 last updated on 20/Mar/21

Center point the circle is (−2,3)  distance from A to center point is (√(5^2 +5^2 )) =5(√2)  radius of circle r =(√(4+9+12)) = 5  so the shortest distance from A to the circle  is 5(√2) −5

$$\mathrm{Center}\:\mathrm{point}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{is}\:\left(−\mathrm{2},\mathrm{3}\right) \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{A}\:\mathrm{to}\:\mathrm{center}\:\mathrm{point}\:\mathrm{is}\:\sqrt{\mathrm{5}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} }\:=\mathrm{5}\sqrt{\mathrm{2}} \\ $$$$\mathrm{radius}\:\mathrm{of}\:\mathrm{circle}\:\mathrm{r}\:=\sqrt{\mathrm{4}+\mathrm{9}+\mathrm{12}}\:=\:\mathrm{5} \\ $$$$\mathrm{so}\:\mathrm{the}\:\mathrm{shortest}\:\mathrm{distance}\:\mathrm{from}\:\mathrm{A}\:\mathrm{to}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\mathrm{is}\:\mathrm{5}\sqrt{\mathrm{2}}\:−\mathrm{5}\: \\ $$

Commented by EDWIN88 last updated on 20/Mar/21

Commented by greg_ed last updated on 20/Mar/21

obviously !

$$\boldsymbol{\mathrm{obviously}}\:!\: \\ $$

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