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1.8. Problems on Quadratics 41

14. (a) Find necessary and sufficient conditions on a, b, w so that the
    roots of z2 + 2u~ + b = 0 and z - w = 0 are collinear in the
    complex plane.
    (b) Find necessary and sufficient conditions on a, b, c, d so that the
    roots of z2 + 2~3% + b = 0 and r2 + 2cr + d = 0 are collinear in
    the complex plane.


15. Show that, if -(b/u)cos2{(1/4)arc cos[(b2 - 8ac)/b2]} exists, then it
    is a root of the equation ax2 + bx + c = 0.


16. Let u and v be the roots of the equation

z + (l/z) = 2(cos 4 + i sin 4) where 0 < 4 < ?r.

(a) Show that U+ i and v + i have the same argument and that u - i

and v - i have the same modulus.

(b) Find the locus of the roots u, v in the complex plane when 4

varies from 0 to 7r.

17. Solve

x2 - (2~ - b - c)x + (a2 + b2 + c2 - bc - cu - ub) = 0.

18. Let a, b, c be nonzero integers such that the greatest common divisor
    of b and UC is 1. Prove that ax2 + bx + c and ax2 + bx - c can both be
    written as the product of linear polynomials with integer coefficients
    if and only if UC = rs(r2 --s2) and b2 = (r2 + s~)~, where r and s are
    relatively prime integers.


19. If a, b and h are constants, prove that the maximum and minimum
    values of a cos2 0 + 2h sin 8 cos 0 + b sin2 0 are the roots of the equation
    (x - u)(x - b) = h2.


20. What conditions must be satisfied by the constants a, b, c for the
    quadratic function

f(x,y)=x2-y2+2ux+2by-c

to be the product of two linear factors?

21. (a) If the line y = mx+c is tangent to the curve b2x2 -a2y2 = a2b2,
    show that u2m2 = b2 + c2.
    (b) Chords of th e circle x2 + y” = r2 touch the hyperbola b2x2 -
    u2y2 = u2b2. Find the equation of the locus of the midpoints.


22. Determine all those quadratic polynomials whose zeros are symmetric
    about the imaginary axis, i.e. r + is is a zero if and only if -r + is is
    a zero.