120 .1 \ ILYTH QXOJfXTBr ("<'' 



8. A lino drawn through the point ( I. ."> ) make* an isosceles triangle 

 with the lines 3z=4y+7 and 5y = 12 r ; : lind its equation. 



9. Prove analytically that the diagonals of a square are of equal 

 length, bisect each other, and are at right angles. 



10. Prove analytically that the line joining the middle points of two 

 sides of a triangle is parallel to the third Hide and equal to half its length. 



11. Find the locus of the vertex of a triangle whose base is 2 a and 

 the difference of the squares of whose sides is 4c*. Trare the locus. 



12. Find the equations of the lines from the vertex (4, 3) of the tri- 

 angle of Ex. 4, trisecting the opposite side. What are the ratios <>i 

 areas of the resulting triangles ? 



13. A point moves so that the sum of its distances from the lines 

 y-Sx-f 11 =0 and 7z-2y+l=0 is 0. Find the equation of its 

 locus. Draw the figure. 



14. Find the equation of the path of the moving point of Ex. 1:5. if 

 the distances from the fixed lines are in the ratio 8:4, 



15. Solve examples 13 and 14, taking the given lines as axes. 



16. The point (2, 9) is the vertex of an isosceles right triangle whose 

 hypotenuse is the line 3z~7y = 2. Find the other vertices of the 

 triangle. 



17. The axes of coordinates being inclined at the angle 60, find tin- 

 equation of a line parallel to the line x + y = 3 a, and at a distance 



~ from it. 



18. Find the point of intersection of the lines 



2a f a 7r\ 



p = 7 and pcos(0-- =a. 



For what value of 6, in each line, is p = co ? At what angles do these lines 

 cut their polar axes? Find the angle between the lines. Plot these 1 



19. Find the equation of a straight line through the intersection of 

 y = 7x - 4 and 2 z -f y = 5, and forming with the ar-axis the an- 



20. Find the equation of the locus of a point which moves so as to be 

 always equidistant from the points (2, 1) and (- :;. -2). 



