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  • The number of points of intersection of <img alt="\mathrm{|z-(4+3 i)|=2\; and \;|z|+|z-4|=6}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%7Cz-%284&plus;3%20i%29%7C%3D2%5C%

The number of points of intersection of \mathrm{|z-(4+3 i)|=2\; and \;|z|+|z-4|=6}, \mathrm{z \in \mathbb{C}}, is

Option: 1

0


Option: 2

1


Option: 3

2


Option: 4

3


Answers (1)

best_answer

\mathrm{|z-(4+3 i)|=2} is a circle with centre: (4,3) and radius =2.
\mathrm{|z|+|z-4|=6} is an ellipse with foci: \mathrm{ (0,0) (4,0), \quad 2 a=6 \Rightarrow a=3, \quad 2 a e=4 \Rightarrow a e=2 \Rightarrow e=\frac{2}{3}}, \\ \mathrm{ \quad b^{2}= a^{2}\left(1-e^{2}\right)=9\left(1-\frac{4}{9}\right)=5 \Rightarrow b=\sqrt{5}.}

So there are 2 points of intersection 

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Nehul

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