If A 0-0- B cos- sin- and C cos- sin- represent the vertices of a right angled triangle- then which of the following is-are correct-A- sin-2-1-2B- cos-2-1-2C- Coordinates of orthocentre are 0-0D- Coordinates of orthocentre are cos- sin-

The correct options are A sin(α β2) = 1√(2) B cos(α β2) = 1√(2) C Coordinates of orthocentre are (0, 0)Given : A(0,0), B(cosα,sinα) and C(cosβ,sinβ) Findin ...

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Standard XII

Mathematics

Orthocentre

If A 0,0, B c...

Question

If $A (0, 0), B (cos α, sin α)$ and $C (cos β, sin β)$ represent the vertices of a right angled triangle, then which of the following is/are correct?

sin (\frac{α - β}{2}) = \frac{1}{\sqrt{2}}

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cos (\frac{α - β}{2}) = - \frac{1}{\sqrt{2}}

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Coordinates of orthocentre are

(0, 0)

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Coordinates of orthocentre are

(cos α, sin α)

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Solution

The correct options are
A $sin (\frac{α - β}{2}) = \frac{1}{\sqrt{2}}$
B $cos (\frac{α - β}{2}) = - \frac{1}{\sqrt{2}}$
C Coordinates of orthocentre are $(0, 0)$
Given : $A (0, 0), B (cos α, sin α)$ and $C (cos β, sin β)$
Finding the lengths of the triangle,
$A B = \sqrt{{cos}^{2} α + {sin}^{2} α} = 1 units$
$A C = \sqrt{{cos}^{2} β + {sin}^{2} β} = 1 units$
$B C = \sqrt{(cos α - cos β)^{2} + (sin α - sin β)^{2}} \Rightarrow B C = \sqrt{2 - 2 (cos α cos β + sin α sin β)} \Rightarrow B C = \sqrt{2 - 2 cos (α - β)} units$

As $A C = A B$ , triangle is right angled isosceles triangle. So, hypotenuse is $B C$
Now, using Pythagoras theorem, we get
$A B^{2} + A C^{2} = B C^{2}$
$\Rightarrow 2 = 2 - 2 cos (α - β)$
$\Rightarrow cos (α - β) = 0 \Rightarrow α - β = \frac{(2 n + 1) π}{2}, n \in Z ∴ sin (\frac{α - β}{2}) = cos (\frac{α - β}{2}) = \pm \frac{1}{\sqrt{2}}$

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

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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?

If $A (0, 0), B (cos α, sin α)$ and $C (cos β, sin β)$ represent the vertices of a right angled triangle, then which of the following is/are correct?