P and Q are two points on a circle of centre C and radius α, the angle PCQ being 2 θ then the radius of the circle inscribed the triangle CPQ is maximum when -

Question

P and Q are two points on a circle of centre C and radius α, the angle PCQ being 2 θ then the radius of the circle inscribed the triangle CPQ is maximum when - (A)  sin θ=(√3-1/2 √2) (B)  sin θ=(√5-1/2) (C)  sin θ=(√5+1/2) (D)  sin θ=(√5-1/4)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/fc4edccc8c9f94afdab451c27e7e2ff4-.png" /> s =(2 α+2 α sin θ/2)=α+α sin θ Δ=(1/2) α2 sin 2 θ r =(Δ/ s )=(1/2) α ( sin 2 θ/1+ sin θ) Maximize 'r'.

Q. $P$ and $Q$ are two points on a circle of centre $C$ and radius $α$ , the angle $PCQ$ being $2 θ$ then the radius of the circle inscribed the triangle $CPQ$ is maximum when -

$sin θ = \frac{3 - 1}{2 2}$

$sin θ = \frac{5 - 1}{2}$

$sin θ = \frac{5 + 1}{2}$

$sin θ = \frac{5 - 1}{4}$

$s = \frac{2 α + 2 α s i n θ}{2} = α + α sin θ$
$Δ = \frac{1}{2} α^{2} sin 2 θ$
$r = \frac{Δ}{s} = \frac{1}{2} α {\frac{s i n 2 θ}{1 + s i n θ}}$
Maximize ' $r$ '.

Q. P and Q are two points on a circle of centre C and radius α, the angle PCQ being 2θ then the radius of the circle inscribed the triangle CPQ is maximum when -

sinθ=22​3​−1​

sinθ=25​−1​

sinθ=25​+1​

sinθ=45​−1​