Level – I
1. Prove that the points (4 , -1), (6 , 0), (7 , 2) and (5 , 1) are the vertices of rhombus.
2. Find all points on x + y = 4 that lie at a unit distance from the line 4x + 3y – 10 = 0.
3. Find the distance of the line 3x + y + 4= 0 from the point (2, 5) measured parallel to the line 3x – 4y + 8 = 0.
4. Prove that all the vertices of an equilateral triangle can not be integral points (an integral point is a point both of whose coordinates are integers).
5. A line 4x + y =1 through the point A(2 , -7) meets the line BC whose equation is 3x – 4y + 1 = 0 at the point B. Find the equation to the line AC so that AB = AC.
6. Find α, if (α, α2) lies inside the triangle having sides along the lines 2x + 3y = 1 , x + 2y – 3 = 0, 6y = 5x – 1.
7. If points A (5/√2 ,√3 ) and B (cos2θ, cos θ) are on the same side of the line 2x – y = 1, then find the values of q in [ p , 2p].
8. Let (h, k) be a fixed point where h, k > 0. A straight line passing through this point cuts the positive direction of coordinate axes at points P and Q. Find the minimum area of the ΔOPQ, O being the origin.
9. If through the angular points of a triangle straight lines be drawn parallel to the sides and if the intersections of these lines be joined to the opposite vertex of the triangle. Show that the joining lines so obtained will meet in a point.
10. The points (1, 3) and (5, 1) are two opposite vertices of a rectangle. The other two vertices lie on the line y = 2x + c. Find c and the remaining two vertices.