First slide

Definition of a circle

Question

Let $A \equiv (- 4, 0) and B \equiv (4, 0)$ If the number of points on the circle $x^{2} + y^{2} = 16$ such that the area of the triangle whose vertices are A, B, and C is a positive integer N then the value of N is __________

Moderate

Solution

Required area
$A = \frac{1}{2} \times 8 \times 4 sinθ = 16 sinθ$
Now, the area is an integer. Then the possible values of sin $θ$ are
$\frac{1}{16}, \frac{2}{16}, \dots, \frac{15}{16}$
i.e., 15 points in each quadrant or 60 + 2 more with sin $θ$ =1
Therefore, N=62

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