[If a is the angle which the line joining (x, y) and (x,., y,.) makes
with y = 0, e iia = (x,. + iy,.—x-iy)/(x r — iy,.-x + iy).]

Ex. 7. The tangents from the real foci of 2' = to any curve of the
tangential pencil 2 + fc2' = touch a curve confocal with 2 = 0.
Ex. 8. The sum of the angles made with a fixed line by the tangents
to a curve of class m from any point of an m-ic through all its foci
is constant.
The asymptotes of the latter curve are parallel to the sides of a
regular polygon.
[See Ex. 6, and § 1, Ex. 4.]

Ex. 9. If f(x) = is an algebraic equation of degree n with roots
»! + j|3j , (X 2 + i/3 2 , . . . , and the roots of /' (x) = are OL\ + t , Oi\ + iji\ , . . . ,
prove that the centroid of the points (pt lt ft), (0C it j3 2 ), ... coincides
with the centroid of the points (a'D/3'j), (a' 2 , /3' 2 )

Show that there exists a curve of class n — 1 with real foci at the
former points which touches at its middle point the line joining any
two of the latter points.
[The curve is (Xa 1 + ^ 1 + l)- I + (\a 2 + /i /3 2 + l)- 1 + ... = 0.]

Ex. 10. Fixed segments AiB u A^B^, ..., A m B m subtend angles at P
whose sum a- is constant. Show that the locus of P is a m-circular Zw-ic.
