Given a triangle ABC with circumcentre at origin. If a $\leq$ 4 sin A then for any point (a, y) inside the circumcircle of $\Delta$ABC
- $|xy| < 2$
- $2 < |xy| < 4$
- $\dfrac{1}{2}$ < |xy| < $\dfrac{3}{4}$
- $|xy| > 4$
Using the sine rule, the side length a is given by 2R sin A, where R is the circumradius. The condition a <= 4 sin A simplifies to 2R sin A <= 4 sin A, which implies R <= 2. Since the circumcentre is at the origin, the equation of the circumcircle is x^2 + y^2 = R^2 <= 4. For any point (x, y) inside this circle, we have x^2 + y^2 < 4, and by the AM-GM inequality, |xy| <= (x^2 + y^2)/2 < 2.
The problem asks for the bound of the product |xy| for a point (x, y) inside a circumcircle centered at the origin with radius R. Using the extended sine rule, a is less than or equal to 4 sin A implies the circumradius R must be 2. Since the point lies strictly inside this circumcircle, its coordinates must satisfy x squared plus y squared is strictly less than R squared, which is 4. We know that 2|xy| is always less than or equal to x squared plus y squared by the AM-GM inequality, meaning 2|xy| is less than 4. Therefore, the product |xy| must be less than 2.