Assignment: Straight Line

1. If the vertices of a triangle have integral coordinates, prove that the triangle cannot be equilateral.

2. Let be the vertices of a , where are the roots of the equation

; are the roots of equation and are the roots of equation

, being positive. Find and the centroid of .

3. Let be the roots of the equation and be those of . If the four points be such that the sum of the ratios in which C and D divide AB is zero, prove that

4. If the axes are shifted to the point (1, -2) without rotation, what do the following equations become ?

(i) (ii)

5. Prove that the triangle which has one of the angles as , cannot have all vertices with integral coordinates.

6. If the points (4, 7) and , lie on the same side of the line , then prove that lies in the first quadrant.

7. Find the values of so that the point lies on or inside the triangle having the sides

8. Find all the values of for which the point lies inside the square formed by the lines and

9. Let be a point and let OAB be a triangle with vertices

. Find is P lies inside the triangle OAB.

10. The equations of the two straight lines through (7, 9) and making an angle of with the line

11. Given ‘n’ straight lines and a fixed point O, a straight line is drawn through O meeting lines in the points

and on it a point is taken such that . Find the locus of R

12. Let (h, k) be a fixed point where . A straight line passing through this point cuts the positive direction of the co-ordinate axes the points P and Q. Find the minimum area of the ; O being the origin.

13. Consider the point A(3, 4) and B(7, 13). If ‘P’ be a point of the line such that is minimum, Find the co-ordinates of ‘P’.

14. Consider the point A(0, 1) and B(2, 0) ‘P’ be a point on the line . Find the coordinates of the point ‘P’ such that is maximum.

15. Triangle ABC is right angled at B. If and side AC slides along the positive co-ordinate axes in such a way that ‘B’ always remains in the first quadrant and opposite side of Origin w.r.t. AC, Find the locus of B.

16. An equilateral triangle ABC of side length 2 units. Locus of vertex C as the side AB slides along the positive coordinate axes and C lies opposite side of origin w.r.t. AB

17. Abscissae and ordinates of ‘n’ given points are in A.P. with first term ‘a’ and common difference 1 and 2 respectively. If algebraic sum of perpendiculars from these given points on a variable line which always passes

through the point is zero. Find the values of ‘a’ and ‘n’