Unknown

42 1. Fundamentals

23. Find the equations of those conjugate diameters of the ellipse b2x2 +
    U2Y2 = a2b2 which are of equal length. (Two diameters are conjugate
    if each is the locus of midpoints of chords parallel to the other. Refer
    to Exercise 2.16.)


24. Let ax2 + bx + c be a quadratic polynomial with real coefficients for
    which hrz2 + bz + cl 5 1 for 0 5 2 5 1. Prove that Ial + Ibl + ICI 5 17.
    Give an example for which equality holds.

1.9 Other Problems

1. Suppose that t3 + pt + q = 0 has a nonreal root a + bi, where a, b, p,
   q are all real and q # 0. Show that aq > 0.


2. Consider a polynomial f(z) with real coefficients having the property
   fb(x)) = s(f(+)) f or every polynomial g(x) with real coefficients.
   Determine and prove the nature of f(x).


3. If a, b, c, d are real numbers, show that each of the two systems of
   three equations is equivalent to the other:

I. a2+b2 = 2 c2 +d2 = 2 ac = bd

II. a2+c2 = 2 b2 +d2 = 2 ab= cd.

4. Find a simple expression for the positive root of

x3-3x2-Z:-dLo0.

5. Show that any root of

(x + a + b)(x-’ + u-l + b-‘) 7 1

is a root of

(x” + a” + b”)(x-” + a-” + b-“) = 1,

where n is any odd integer and where a and b are both different from

0.

6. (a) Given that x + a + &q = 0, where x is not 0, verify that

(b) Given that y = pz + q, where p 2 0 and 2 + a + m

verify that

y + (up - q) + &P - q>2 - (bp2 - 2apq + q”) = 0.

= 0,