Vertices of a variable triangle are (3$$ , $$4) (5cos$$θ,$$5sin$$θ$$) and (5sin$$θ,−$$5cos$$θ$$) . Then locus of its orthocenter is

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Vertices of a variable triangle are (3$$ , $$4) (5cos$$θ,$$5sin$$θ$$)      and (5sin$$θ,−$$5cos$$θ$$)     . Then locus of its orthocenter is

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Circumcentre of the triangle is (0$$ , $$0) and  Centroid (3+5cos$$θ$$+5sin$$θ$$3$$,$$4+5sin$$θ−$$5cos$$θ$$3)                                                                     [∵  Centroid divides the join of orthocenter and circumentre in $$2$$ : $$1$$ ratio ⇒ $$G=2S+O3$$ If $$S=(0$$,$$0)$$ then $$3G=Olet$$ (h$$ , $$k) represents orthocenter and  $$3G=3+5cos$$θ$$+5sin$$θ ,$$4+5sin$$θ$$-5cos$$θ        ∴$$h=3+5cos$$θ$$+5sin$$θ           $$k=4+5sin$$θ−$$5cos$$θ   Adding and subtracting the above equations we get                           ⇒$$sin$$θ$$=h+k$$−$$710$$;$$cos$$θ$$=h$$−$$k+110$$ squarring and adding                                                                                                            ⇒$$(h+k$$−$$7)2+(h$$−$$k+1)2=100$$                                                                                       ∴  Locus of orthocenter is   $$(x+y$$−$$7)2+(x$$−$$y+1)2=100$$

Vertices of a variable triangle are (3 , 4) (5cosθ,5sinθ) and (5sinθ,−5cosθ) . Then locus of its orthocenter is

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