The vertices of a variable triangle are $$3$$,$$4$$, $$5cos$$θ,$$5sin$$θ, and $$5cos$$θ,−$$5sin$$θ, where θ∈R. The locus of its orthocenter is

Question

The vertices of a variable triangle are $$3$$,$$4$$, $$5cos$$θ,$$5sin$$θ, and $$5cos$$θ,−$$5sin$$θ, where θ∈R. The locus of its orthocenter is

Infinity Learn · Accepted Answer

Distance of all the points from (0$$,$$0) are $$8$$ units. The means the circumcenter of the triangle formed by the given points is (0$$,$$0) . If G≡h,k is the centroid of the triangle, then $$3h=3+5cos$$θ$$+$$$$sin$$θ. $$3k=4+5sin$$θ−$$cos$$θ. If Hα,β is the orthocente, then OG:$$GH=1$$:$$2$$ or α$$=3h$$, β$$=3kcos$$θ$$+$$$$sin$$θ$$=$$α−$$35$$.$$sin$$θ−$$cos$$θ$$=$$β−$$45or$$ $$sin$$θ$$=$$α$$+$$β−$$710$$,$$cos$$θ$$=$$α−β$$+110Thus$$, the locus of α,β is $$x+y$$−$$72+x$$−$$y+12=100$$

The vertices of a variable triangle are $(3, 4)$ , $(5 cosθ, 5 sinθ),$ and $(5 cosθ, - 5 sinθ),$ where $θ \in R .$ The locus of its orthocenter is

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The vertices of a variable triangle are 3,4, 5cosθ,5sinθ, and 5cosθ,−5sinθ, where θ∈R. The locus of its orthocenter is

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The vertices of a variable triangle are $(3, 4)$ , $(5 cosθ, 5 sinθ),$ and $(5 cosθ, - 5 sinθ),$ where $θ \in R .$ The locus of its orthocenter is