Circles
Circle
Allen Star Batch
Grade 11
Question:
Let $PT$ be a tangent from the point $P(5,3 + \sqrt{3})$ to the circle $x^2 + y^2 + 4x - 6y - 3 = 0$, with centre $C$, at $T$ and $AB$ is a secant which passes through $P$ such that $BT$ is the normal at $T$. If $Ar(\triangle CAB) + Ar(\triangle CAT) = \frac{k}{25}$, then find the value of $(\sqrt{k} - 15)$ ([.] denotes G.I.F.).
Step-by-Step Solution
Key Concept: Use the tangent half-angle formula to relate the triangle dimensions and compute the combined area using sine addition.
Given $CT = CB = r = 4$, we calculate $PT = \sqrt{5^2 + (3\sqrt{3})^2 + 4 \times 5 - 6(3\sqrt{3}) - 3} = 6$. Setting $\angle ACB = \alpha$, the areas of triangles $ACB$ and $CAT$ are $8\sin\alpha$ and $8\sin(\pi - \alpha)$ respectively. In triangle $PBT$, $\tan(\alpha/2) = BF/PT = 8/6$, giving $\sin\alpha = 24/25$. Therefore, the total area is $16\sin\alpha = 384/25$, and $\lambda = 384$, so $\sqrt{384} - 15 = 4$.
Correct Answer: 4