An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

58

2x +a —p(y-P)
may be found from the equation x'= - 2
1 q-p"
required a proof.
81. If a perpendicular upon the tangent from the focus
of an ellipse be drawn; prove that the distance of the point of
their intersection from the centre of the curve is invariable.
82. Given the length of the perpendicular from the centre
of an ellipse upon its tangent; to find the point of contact.
83. In an hyperbola, given the length of the perpendicular from either focus upon the tangent; to determine the
point of contact.
84. In the lemniscata of Bernoulli whose equation is
(x2 + y2)2 =2a2 (X2 _ y2), find the equation to the rectilineal
tangent which shall make equal angles with the co-ordinate
axes.
85. Required the equation to the tangent making equal
angles with the axes of a curve whose equation is x3 + y3 = a;
and find where it meets them.
86. In the curve defined by x2y2 = (a2 - _2) (x - b)2, it
is required to find the equations to the tangents inclined to
the axis of x at angles whose trigonometrical tangents are
a2
+ -- 1: and to determine the points of contact.
87. Required the equations to the tangents inclined at
angles of 30~ to the axes of the ellipse defined by the equation
y2 — 2y + 3X2 - 2y - 4x + 5 = 0; and find their points of intersection with the axes.
88. Tangents to an ellipse, from a given point a, /3 without it, may be drawn by means of the equations
a2y2 + - ^2" = a2b and a2cy + b2ax = 2b;
required a proof.
89. It is required to prove that the equation to the
straight line joining the points of contact as determined by
the expressions in the last example is a203y -t b2aXGa'b2