A TANGENT IS DRAWN FROM THE POINT (4,0) TO THE CIRCLE x^2 + y^2 = 8 TOUCHES IT AT A POINT A IN THE FIRST QUADRANT. THE COORDINATE OF POINT B ON THE CIRCLE SUCH THAT AB=4 ARE

1. 2,-2
2. -2,2
3. -2ROOT2,0
4. 0,-2ROOT2

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$$\begin{gathered} {\mathrm{x}}^2+{\mathrm{y}}^2=8={\left(2\sqrt{2}\right)}^2 \\ \mathrm{Equation} \mathrm{of} \mathrm{tangent} \mathrm{to} \mathrm{circle} \mathrm{at} \mathrm{A}\left(2\sqrt{2}\mathrm{cos\theta },2\sqrt{2}\mathrm{sin\theta }\right) \mathrm{is} \\ 2\sqrt{2}\mathrm{xcos\theta }+2\sqrt{2}\mathrm{ysin\theta }=8 \\ \mathrm{It} \mathrm{passes} \mathrm{through} \left(4,0\right) \\ \Rightarrow 8\sqrt{2} \mathrm{cos\theta }+0=8 \\ \Rightarrow \mathrm{cos\theta }=\frac{1}{\sqrt{2}} \\ \mathrm{For} \mathrm{first} \mathrm{quadrant}, \\ \mathrm{\theta }=45 \\ \mathrm{sin\theta }=\frac{1}{\sqrt{2}} \\ \left(2\sqrt{2}\mathrm{cos\theta },2\sqrt{2}\mathrm{sin\theta }\right)=\left(2\sqrt{2}.\frac{1}{\sqrt{2}}2\sqrt{2}.\frac{1}{\sqrt{2}}\right)=\mathrm{A}\left(2,2\right) \\ \mathrm{Let} \mathrm{point} \mathrm{B} \mathrm{be} \mathrm{B}\left(2\sqrt{2}\mathrm{cos\alpha },2\sqrt{2}\mathrm{sin\alpha }\right) \\ \mathrm{AB}=4 \\ {\mathrm{AB}}^2=16 \\ \Rightarrow {\left(2-2\sqrt{2}\mathrm{cos\alpha }\right)}^2+{\left(2-2\sqrt{2}\mathrm{sin\alpha }\right)}^2=16 \\ \Rightarrow 4{\left(1-\sqrt{2}\mathrm{cos\alpha }\right)}^2+4{\left(1-\sqrt{2}\mathrm{sin\alpha }\right)}^2=16 \\ \Rightarrow {\left(1-\sqrt{2}\mathrm{cos\alpha }\right)}^2+{\left(1-\sqrt{2}\mathrm{sin\alpha }\right)}^2=4 \\ \Rightarrow 1+2{\cos }^2\mathrm{\alpha }-2\sqrt{2}\mathrm{cos\alpha }+1+2{\sin }^2\mathrm{\alpha }-2\sqrt{2}\mathrm{sin\alpha }=4 \\ \Rightarrow 2\left({\cos }^2\mathrm{\alpha }+{\sin }^2\mathrm{\alpha }\right)-2\sqrt{2}\left(\mathrm{cos\alpha }+\mathrm{sin\alpha }\right)=4-2 \\ \Rightarrow 2-2\sqrt{2}\left(\mathrm{cos\alpha }+\mathrm{sin\alpha }\right)=2 \\ \Rightarrow -2\sqrt{2}\left(\mathrm{cos\alpha }+\mathrm{sin\alpha }\right)=0 \\ \Rightarrow \mathrm{cos\alpha }=-\mathrm{sin\alpha } \\ \Rightarrow \mathrm{tan\alpha }=-1 \\ \Rightarrow \mathrm{\alpha }=\frac{3\mathrm{\pi }}{4} \mathrm{or} \frac{7\mathrm{\pi }}{4} \\ \mathrm{B}\left(2\sqrt{2}\mathrm{cos\alpha },2\sqrt{2}\mathrm{sin\alpha }\right) \\ =\mathrm{B}\left(2\sqrt{2}\cos \frac{3\mathrm{\pi }}{4},2\sqrt{2}\sin \frac{3\mathrm{\pi }}{4}\right) \mathrm{or} \mathrm{B}\left(2\sqrt{2}\cos \frac{7\mathrm{\pi }}{4},2\sqrt{2}\sin \frac{7\mathrm{\pi }}{4}\right) \\ =\mathrm{B}\left(-2,2\right) \mathrm{or} \mathrm{B}\left(2,-2\right) \end{gathered}$$

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