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LEAVING CERTIFICATE
2. Prove that for all values of m the straight line
y = m (x—a) -f aV\
touches the circle (^—a)2 + ^y2 = «2.
Write down the equations of the two tangents to this
circle which are parallel to the straight line Ay = 3 x, and
the equations of the two tangents which are perpendicular
to this straight line.
Choose one mutually perpendicular pair of these tangents,
and prove that their point of intersection lies on the circle
x* -\- y* — 2 a x = a2.
3. Find the equation of the normal at any point of the
parabola y* = A ax, and prove that the portion of the axis
intercepted between the ordinate and the normal at any
point is constant.
Prove that the chord of the parabola y2 = 10 x, whose
equation is % + 3y = 10, subtends a right angle at the vertex
of the parabola.
4. From a variable point P on the circumference of a
circle a perpendicular PN is drawn to a fixed diameter AB,
and a point Q is taken in PN such that QN = mPN, where m
is a constant.
Prove that the locus of Q is an ellipse, and draw sketches
to show the relation of the ellipse to the circle when m < 1
and when m > 1.
Prove also that the tangent to the circle at P, and that
to the ellipse at Q, intersect on the fixed diameter.
5. Prove that the point whose co-ordinates are ct, c\t,
where t is a variable quantity, lies on a rectangular hyperbola,
and find the equation of the chord joining the points at
which t has the values ^ and
Find the equation of a tangent to the hyperbola which
is parallel to this chord.
If Qx, <2a are the extremities of the chord, P the point
of contact of the parallel tangent, PK, Q-JK-x, the
perpendiculars from P, Qx, Q2 on an asymptote, prove that
QxKx-Q2K2 = PKz.