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# 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.

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
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.

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