Beautiful problems from Coordinate Geometry

The following problems are collected from a variety of Math Olympiads and mathematics contests like I.S.I. and C.M.I. Entrances. They can be solved using elementary coordinate geometry and a bit of ingenuity.

1. The equation \( x^2 y - 3xy + 2y = 3 \) represents:
   • (A) a straight line;
   • (B) a circle;
   • (C) a hyperbola
   • (D) none of the foregoing curves;
2. The equation \( r = 2a \cos \theta + 2b \sin \theta \) in polar coordinates represents:
   • (A) a circle passing through the origin;
   • (B) a circle with the origin lying outside it;
   • (C) a circle with radius \( 2 \sqrt {a^2 + b^2 } \) ;
   • (D) a circle with the center at the origin;
3. The curve whose equation in polar coordinates is \( r \sin^2 \theta - \sin \theta - r = 0 \), is
   • (A) an ellipse;
   • (B) a parabola;
   • (C) a hyperbola;
   • (D) none of the foregoing curves;
4. A point P on the line 3x + 5y = 15 is equidistant from the coordinate axes can lie in
   • (A) quadrant I only;
   • (B) quadrant I or quadrant II;
   • (C) quadrant I or quadrant III;
   • (D) any quadrant;
5. The set of all points (x, y) in the plane satisfying the equation \( 5x^2 y - xy + y = 0 \) forms:
   • (A) A straight line;
   • (B) a parabola;
   • (C) a circle;
   • (D) none of the foregoing curves;
6. The equation of the line through the intersection of the lines $$ 2x + 3y + 4 = 0 \textrm{and} 3x + 4y - 5 = 0 $$ and perpendicular to \( 7x - 5y+ 8 = 0 \) is:
   • (A) 5x + 7y - 1 = 0;
   • (B) 7x + 5y + 1 = 0;
   • (C) 5x - 7y + 1 = 0;
   • (D) 7x - 5y - 1 = 0;
7. The two equal sides of an isosceles triangle are given by the equations y = 7x and y = -x and its third side passes through (1, -10). Then the equation of the third side is
   • (A) 3x + y + 7 = 0 or x - 3y - 31 = 0
   • (B) x + 3 y + 29 = 0 or -3x + y + 13 = 0
   • (C) 3x + y + 7 = 0 or x + 3y + 29 = 0
   • (D) x - 3y - 31 = 0 or - 3x + y + 13 = 0
8. The equations of two adjacent sides of a rhombus are given by y = -x and y = 7x. The diagonals of the rhombus intersect each other at the point (1, 2). The area of the rhombus is:
   • (A) \( \frac{10}{3} \)
   • (B) \( \frac{20}{3} \)
   • (C) \( \frac{50}{3} \)
   • (D) none of the foregoing quantities.

More problems are in the Cheenta student portal. You may send answers to support@cheenta.com.

We will keep on adding more problems in this list as well.