JEE Advanced Maths: Coordinate Geometry / Theory of Equations questions with solutions

Q1. Solve the simultaneous equations: x² + y² + 6x + 2y = 0 and x + y + 8 = 0. Find all real solution pairs (x, y).

1. (x, y) = (-4, -4) and (-6, -2)
2. (x, y) = (-2, -6) and (-6, -2)
3. no real solution
4. (x, y) = (0, -8) and (-8, 0)

Answer: (x, y) = (-4, -4) and (-6, -2)

The first equation is a circle; the second is a line. Substituting the line into the circle gives a quadratic whose discriminant tells how many intersection points exist (here the line is tangent).