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15.
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15.
LEAVING CERTIFICATE EXAMINATION. 1003
2. Define the radical axis of two circles. Prove that the three radical
axes of three circles taken together meet in a point.
From a given point 0 a straight line is drawn to a variable point
P on a fixed circle ABC; on OP as diameter a circle is described ;
if the tangent to this circle at 0 meef the radical axis of the two
circles at Q, find the locus of Q.
3. QA, OB, OC are adjacent edges of a rectangular block of which OD
is a diagonal, prove that OD passes through the centroid of ABC.
Calculate the area of the triangle ABC in terms of OA, OB, OC
(x, y, 4
Alternative Questions.
4 a. Enunciate and prove Menelaus’ theorem.
(L, L'), (M, M'). (N, JV) are pairs of points on BC, CA, AB such
that LB = CL', MC = AM', NA — BN'; if L, M, N lie in a
straight line, prove thao L', M', N' also lie in a straight line.
Or
46. Prove that if a straight line be divided into two parts, the
rectangle contained by the two parts is a maximum when the
parts are equal.
ABC is a triangle and P is a point in BC; from P parallels PQ,
PR are drawn to BA, CA, meeting AC, AB, in Q, R ; find the
position of P which corresponds to the maximum area of the
parallelogram PQAR.
5a. If A, B be harmonic conjugates with respect to C, D, prove that
C, D are harmonic conjugates with respect to A, B.
Prove that each diagonal of a complete quadrilateral is divided
harmonically by the other diagonals.
Or
56. Prove that the inverse of a circle is either a straight line or a
circle, according to the position of the centre of inversion.
Two unequal circles being given, find the locus of the centre of
inversion in order that each circle may be inverted into a circle
equal in area to the other circle.
ба. Define pole and polar.
A is a fixed point and 0 is a fixed circle, find two points B and C,
such that the triangle ABC may be self-con jugate ; that is, that
each vertex may be the pole of the opposite side.
If PQR be a self-conjugate triangle, find the centre and radius of
the circle with respect to which it is self-conjugate.
Or
бб. ABC is a triangle, P is any point in AB, and a point Q is taken
in AC such that CQ = BP; prove that the radical axis of the
circles circumscribing the triangles ABC, APQ is a fixed line
through A.
15.