Circles
Circle
star_batch_jee_advanced_2025
Grade 11
Question:
On the side $AC$ of an acute angled triangle $ABC$ a point $D$ is taken, such that $AD = 1, DC = 2$ and $BD$ is an altitude of $\triangle ABC$. A circle of radius $2$, which passes through points $A$ and $D$ and touches a circle at the point $D$ circumscribed about the $\triangle BDC$. If the area of $\triangle ABC$ is $A$ then the value of $\frac{1}{11}[A]$ is equal to __________.
(Where $[.]$ represents G.I.F)
Step-by-Step Solution
Key Concept: The inscribed angle theorem and chord length formula relate trigonometric angles to geometric distances.
From triangle $AOE$, $\cos \alpha = \frac{1}{2} - \frac{1}{4}$. The angle $\theta = \pi - 2\alpha$, and using the inscribed angle relation, $BD = 2\cot\left(\frac{\pi}{2} - \alpha\right) = 2\tan \alpha = 2\sqrt{15}$. The triangle area is $\frac{1}{2} \cdot AC \cdot BD = \frac{1}{2} \cdot 3 \cdot 2\sqrt{15} = 3\sqrt{15}$ or equivalently $\sqrt{135}$.
Correct Answer: 1