CONIC SECTIONS, ANALYTICAL. 1<~, 



531. If the straight lines represented by the equation 



x f (tan 1 < 4- cos* <f>) - 2xy tan < + \f sin' < = 

 make angles a, /? with the axis of x, 



tan a -tan/? =2. 



532. Form the equation of the straight lines joining the 

 origin to the points given by the equations 



and prove that they will be at right angles if V + k* = c". 



533. The locus of the equation 



*-! * f -l 



y 9 4. _ _ 

 2+ 2+ ... to oo 



is the part of two straight lines at right angles to each other 

 which include one quadrant. 



534. If the formula? for effecting any transformation of co- 

 ordinates be 



x = aX + 6 1" 4- c, y a'X + b' T + c', 



tlit n will (ab - a'b 1 ) (ab' - ab) = bb'- aa. 



'. The expression 



cat? + by* + c + 2ay + 26* f L' 

 is transformed to 



AX' + J17 9 + C+'2.i'.\' IKY + W 

 - -rigin being unaltered ; prove that 





in g sin'U 



being the angles between the axes. 



