In the circle given below, let O A = 1 unit, O B = 13 unit and P Q ⊥ O B . Then, the area of the triangle P Q B (in square units) is :

Question

In the circle given below, let O A = 1 unit, O B = 13 unit and P Q &#8869; O B . Then, the area of the triangle P Q B (in square units) is :

Aicharya · Accepted Answer

Since x-axis is normal to the circle at $O$ , and normal to the circle passes through its centre. So, $O B$ is a diameter.

Now, perpendicular from centre of circle to chord bisects the chord, so $P A = A Q = λ (L e t)$

Also, product of segments of chords in a circle is equal i.e. $O A \cdot A B = A P \cdot A Q$

$\Rightarrow 1 \cdot 12 = λ \cdot λ$

$\Rightarrow λ = 2 \sqrt{3}$ units

Now, area of $Δ P Q B = \frac{1}{2} \times P Q \times A B = \frac{1}{2} \times 2 λ \times A B$

i.e. $Δ = \frac{1}{2} \cdot 4 \sqrt{3} \times 12$ sq. units

$= 24 \sqrt{3}$ sq. units

In the circle given below, let O A = 1 unit, O B = 13 unit and P Q ⊥ O B . Then, the area of the…