Vertices of a variable triangle are (3, 4), (5 cos θ, 5 sin θ) and (5 sin θ, -5 cos θ) where θ∈ R.Then locus of it's orthocenter is
A
+ = 100
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B
+ = 100
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C
+ = 100
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D
+ = 100
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Solution
The correct option is D + = 100 Distance of all the points from (0, 0) are 5 unit. That means circumcentre of the triangle formed by the given point is (0, 0). If G = (h, k) be the centroid of triangle, then 3h=3+5(cosθ+sinθ),3k=4+5(sinθ−cosθ) If H(α,β) be the orthocenter, then OG:GH=1:2⇒α=3h,β=3kcosθ+sinθ=α−35,sinθ−cosθ=β−45⇒sinθ=α+β−710,cosθ=α−β+110Thus,locusof(α,β)is(x+y−7)2+(x−y+1)2=100 Hence, (d) is the correct answer.