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Question-138829




Question Number 138829 by 676597498 last updated on 18/Apr/21
Answered by physicstutes last updated on 19/Apr/21
T: z→ω ⇒ ω = 3z + 2−5i  (a) This transformation is an enlargment of scale factor 3 followed by   a translation of (2,−5).  (b) For invariant point, f(z)=z  ⇒   z = 3z+ 2−5i  ⇒ z = −1+ (5/2)i  (c) a^� = 3a + 2−5i   .....(i)        b^�  = 3b + 2−5i    .....(ii)  (ii)−(i) ⇒  ∣b^� −a^� ∣ = 3∣b−a∣ ⇒ T is a simultitude of radius 3.  (d) if z is a circle centered O and radius 2 then  ∣z∣ = 2   ∣ω−(2−5i)∣=6  , then ω will be a circle centred (2,−5) and radius 6.  since T is an enlargment(scale factor 3) and a translation by (2,−5).
$${T}:\:{z}\rightarrow\omega\:\Rightarrow\:\omega\:=\:\mathrm{3}{z}\:+\:\mathrm{2}−\mathrm{5}{i} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{This}\:\mathrm{transformation}\:\mathrm{is}\:\mathrm{an}\:\mathrm{enlargment}\:\mathrm{of}\:\mathrm{scale}\:\mathrm{factor}\:\mathrm{3}\:\mathrm{followed}\:\mathrm{by}\: \\ $$$$\mathrm{a}\:\mathrm{translation}\:\mathrm{of}\:\left(\mathrm{2},−\mathrm{5}\right). \\ $$$$\left(\mathrm{b}\right)\:\mathrm{For}\:\mathrm{invariant}\:\mathrm{point},\:{f}\left({z}\right)={z} \\ $$$$\Rightarrow\:\:\:{z}\:=\:\mathrm{3}{z}+\:\mathrm{2}−\mathrm{5}{i}\:\:\Rightarrow\:{z}\:=\:−\mathrm{1}+\:\frac{\mathrm{5}}{\mathrm{2}}{i} \\ $$$$\left(\mathrm{c}\right)\:\bar {{a}}=\:\mathrm{3}{a}\:+\:\mathrm{2}−\mathrm{5}{i}\:\:\:…..\left({i}\right) \\ $$$$\:\:\:\:\:\:\bar {{b}}\:=\:\mathrm{3}{b}\:+\:\mathrm{2}−\mathrm{5}{i}\:\:\:\:…..\left({ii}\right) \\ $$$$\left({ii}\right)−\left({i}\right)\:\Rightarrow\:\:\mid\bar {{b}}−\bar {{a}}\mid\:=\:\mathrm{3}\mid{b}−{a}\mid\:\Rightarrow\:{T}\:\mathrm{is}\:\mathrm{a}\:\mathrm{simultitude}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{3}. \\ $$$$\left(\mathrm{d}\right)\:\mathrm{if}\:{z}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{centered}\:\mathrm{O}\:\mathrm{and}\:\mathrm{radius}\:\mathrm{2}\:\mathrm{then}\:\:\mid{z}\mid\:=\:\mathrm{2} \\ $$$$\:\mid\omega−\left(\mathrm{2}−\mathrm{5}{i}\right)\mid=\mathrm{6}\:\:,\:\mathrm{then}\:\omega\:\mathrm{will}\:\mathrm{be}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{centred}\:\left(\mathrm{2},−\mathrm{5}\right)\:\mathrm{and}\:\mathrm{radius}\:\mathrm{6}. \\ $$$$\mathrm{since}\:{T}\:\mathrm{is}\:\mathrm{an}\:\mathrm{enlargment}\left(\mathrm{scale}\:\mathrm{factor}\:\mathrm{3}\right)\:\mathrm{and}\:\mathrm{a}\:\mathrm{translation}\:\mathrm{by}\:\left(\mathrm{2},−\mathrm{5}\right). \\ $$

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