Discussion

Question:

In the xy-plane let A = (-2,0), B = (2,0). Define the set S as the collection of all points C on the circle x² + y² = 4 such that the area of the triangle ABC is an integer. The number of points in the set S is

Explanation:

AB = 4

Based on the figure, area of triangel ABC = 1/2 × AB × |y| = 2|y|

∴ 2|y| must be an integer.

Hence, y can be

Case 1: y = ± 1/2
⇒ x² + (1/2)² = 4
⇒ x² = 15/4
⇒ x = ± √15/2
∴ 4 points i.e, (√15/2 ,1/2), (-√15/2 ,1/2), (√15/2 ,-1/2) & (-√15/2 ,-1/2)

Case 2: y = ± 1
⇒ x² + (1)² = 4
⇒ x² = 3
⇒ x = ± √3
∴ 4 points i.e, (√3 ,1), (-√3 ,1), (√3 ,-1) & (-√3 ,-1)

Case 1: y = ± 3/2
⇒ x² + (3/2)² = 4
⇒ x² = 7/4
⇒ x = ± √7/2
∴ 4 points i.e, (√7/2 ,3/2), (-√7/2 ,3/2), (√7/2 ,-3/2) & (-√7/2 ,-3/2)

Case 1: y = ± 2
⇒ x² + (2)² = 4
⇒ x² = 0
⇒ x = 0
∴ 2 points i.e, (0 ,2) & (0 ,-2)

∴ Total 4 + 4 + 4 + 2 = 14 points

Hence, 14.

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