ANALYTIC GBOMETliY [Cii. XI. 



8. The portions of any .-'noi-l .f an livp-rhohi intercepted between 

 the curve and its conjugate are equal. 



M .,<,) -.HUN. hi.i\\ .1 lan-.-nt parallrl to tin- lim- in .juotinn. 



9. Tli.- ......nliuates of a point are a tan (6 + a) and 6 tan (0 + ft) ; 



that the locus of the point, as varies, is an hyperbola. 



10. Prove that the asymptotes of the hyperbola xy = /are 



x = 5 and y 



11. If the coordinate axes are inclined at an angle o>, find the 

 of an hyperbola whose asymptotes are the lines x = 2 and y = 



respectively, and which passe> tlncu-li tin- point (2, 1). 



12. Find the coordinates of the points of contact of the < -..111111011 

 tangents to the hyperbolas, 



x* - y* = 3 a 2 , and xy = 2 a a . 



13. If a right-angled triangle be inscribed in a rectangular hyperbola, 

 prove that the tangent at the right angle is p*>i]><'mli< ular to the 

 h\Iothenuse. 



14. Show that the line y = mx + 2JV- m always touches the hyper- 



bola xy = * 2 ; and that its point of contact is ( , k^/~^~m \ 



W- m / 



15. Kind the point of the rectangular hyperbola xy = 1'J fr \\hirh 

 the subtangent is 4. Find the subnormal for the same point. 



16. Find the polar of the point (5, 3) on the hyperbola x 2 - 2;/ 2 = 7, 

 \\ith respect to the conjugate hyperbola. Show that this line is tai 



to the given hyperbola, at the other end of the diameter from ( 



17. If an ellipse and hyperbola have the same foci, they intersect at 

 ri-ht angles. 



18. Find tangents to the hyperbola 2y a - 16 x* = 1 which are perp.-n- 

 dicular to its asymptotes. 



19. Find normals to the hyperbola (* "" 3 )* - f- v ~ 2 ^ = 1 which III 

 parallel to its asymptotes. Find the polar of their point of inters*-. 



20. Show that, in an equilateral hyperbola, conjugate diameters are 

 equally inclined to the asymptotes. 



21. Show that two conjugate diameters of a rectangular hyperbola 

 are equal. 



