JEE Advanced Maths: Coordinate Geometry questions with solutions

Q1. Find the area (in square units) of the region enclosed by the curve defined by |x + y| + |x - y| = 11.

1. 121
2. 144
3. 100
4. 169

Answer: 121

The relation |x + y| + |x - y| = 2*max(|x|, |y|) = 11 gives max(|x|, |y|) = 5.5, which is a square with side 11 centred at the origin. Its area is 11² = 121.

Q2. How many lattice points (points with both coordinates as integers) lie strictly inside the triangle with vertices at (0,0), (0,41), and (41,0)?

1. 820
2. 780
3. 901
4. 861

Answer: 780

Interior lattice points satisfy i >= 1, j >= 1, i + j <= 40. For each i from 1 to 39, j ranges from 1 to 40-i, giving (40-i) points. Total = sum_(i=1)³⁹(40-i) = sumₖ₌₁³⁹ k = 39*40/2 = 780.