12 1. Fundamentals(Let the equation of a typical chord be y = mx + k, where k is
a parameter. The midpoint of the chord is given by ((21 + x2)/2,
(yi + y2)/2) where (xi, yi) (i = 1,2) are the endpoints of the chord.
The xi are found by solving the system consisting of the equations
of the chord and the ellipse; eliminating y yields a quadratic in x.
However, it is not necessary to actually determine the xi individually.)- Let a, b be two nonnegative real numbers. Use the fact that the zeros
of the quadratic (t - &)(t - 4) are real to establish the arithmetic-
geometric mean inequality (ab) Ii2 < $(a + b) with equality if and
only if a = b. - An interesting question in numerical approximation is how closely a
nonrational root of an equation can be approximated by a rational.
In this exercise, we see that if a quadratic equation with integer co-
efficients has a nonrational root r, then no rational number can be
any closer to it than the reciprocal of the square of its denominator
multiplied by a constant.
Suppose that a, b, c are integers and that r is a nonrational root of
the quadratic equation at2 + bt + c = 0. Let u = p/q be any rational
number, and suppose that Iu - rI < 1.
Prove that
l/q2 5 IP( 5 1~ - rlK
where K = Z(arj + Ial + lbl.
Deduce that there is a constant M such that
jr -p/q! 1 M/q2 for any rational p/q.Explorations
E.4. Graphical Solution of the Quadratic. Suppose a quadratic equa-
tion x2 - ux + v = 0, with real coefficients and real roots is given. How can
this equation be solved graphically? In other words, segments of length u
and v are given and it is required to use them in determining points in the
plane from which the roots might be found using the ancient Greek tools,
ruler and compasses.
One such method is attributed to Thomas Carlyle. Assume for conve-
nience, that u and v are positive. Construct the circle with the segment
joining (0,l) and ( IJ , v ) as d iameter. Verify that the abscissae of its points
of intersection with the x-axis are the required roots. Relate the condition
that the circle intersects the x-axis to the discriminant condition for real
roots.
Can you find other methods?