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Question Number 170458 by MathsFan last updated on 24/May/22

If the line y=mx+c is a tangent of  a circle x^2 +y^2 =r^2 . Show that,    c^2 =r^2 (1+m^2 )

$${If}\:{the}\:{line}\:{y}={mx}+{c}\:{is}\:{a}\:{tangent}\:{of} \\ $$$${a}\:{circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={r}^{\mathrm{2}} .\:{Show}\:{that},\: \\ $$$$\:{c}^{\mathrm{2}} ={r}^{\mathrm{2}} \left(\mathrm{1}+{m}^{\mathrm{2}} \right) \\ $$

Answered by greougoury555 last updated on 24/May/22

⇒r= ((∣c∣)/( (√(1+m^2 ))))  ⇒r^(2 ) = (c^2 /(1+m^2 ))  ⇒c^2 = (1+m^2 )r^2

$$\Rightarrow{r}=\:\frac{\mid{c}\mid}{\:\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }} \\ $$$$\Rightarrow{r}^{\mathrm{2}\:} =\:\frac{{c}^{\mathrm{2}} }{\mathrm{1}+{m}^{\mathrm{2}} } \\ $$$$\Rightarrow{c}^{\mathrm{2}} =\:\left(\mathrm{1}+{m}^{\mathrm{2}} \right){r}^{\mathrm{2}} \\ $$$$ \\ $$

Commented by MathsFan last updated on 24/May/22

thank you sir

$${thank}\:{you}\:{sir} \\ $$

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