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Question

If A(0,0),B(cosα,sinα) and C(cosβ,sinβ) represent the vertices of a right angled triangle, then which of the following is/are correct?

A
sin(αβ2)=12
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B
cos(αβ2)=12
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C
Coordinates of orthocentre are (0,0)
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D
Coordinates of orthocentre are (cosα,sinα)
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Solution

The correct options are
A sin(αβ2)=12
B cos(αβ2)=12
C Coordinates of orthocentre are (0,0)
Given : A(0,0),B(cosα,sinα) and C(cosβ,sinβ)
Finding the lengths of the triangle,
AB=cos2α+sin2α=1 units
AC=cos2β+sin2β=1 units
BC=(cosαcosβ)2+(sinαsinβ)2BC=22(cosαcosβ+sinαsinβ)BC=22cos(αβ) units

As AC=AB, triangle is right angled isosceles triangle. So, hypotenuse is BC
Now, using Pythagoras theorem, we get
AB2+AC2=BC2
2=22cos(αβ)
cos(αβ)=0αβ=(2n+1)π2,nZsin(αβ2)=cos(αβ2)=±12

ABC is right angled at A, so the coordinates of orthocentre are same as coordinates of A, i.e., (0,0)

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