Advanced High-School Mathematics

SECTION 3.6 Cubic Discriminant 167

x

y

P(a,0)

R

Q(b,0)

x + y = r

2 22

Equation 1: x²+y²=6

7. Above is depicted the circle whose equation is x^2 +y^2 = r^2 , as
   well as the tangent line to this circle at the pointR. The point
   P =P(a,0) is the intersection of this tangent line with thex-axis
   and the pointQ=Q(b,0) as the samex-coordinate as the point
   R.

(a) Using a discriminant argument show that ifmis the slope of

the tangent line, then

m^2 =

r^2

a^2 −r^2

Use this to show thatb=r^2 /a.

(b) Using the Secant-Tangent Theorem (see page 32), give another

proof of the fact thatb=r^2 /a. Which is easier?

3.6 The Discriminant of a Cubic

The the quadratic f(x) = ax^2 +bx+c has associated with it the
discriminantD=b^2 − 4 ac, which in turn elucidates the nature of the
zeros off(x). In turn, this information gives very helpful information
about tangency which in turn can be applied to constrained extrema