Tangents and roots

Is it true to say that all tangents have equal roots? I have a question here that says find the values of the constant, c, for which the straight line y=c-3x is a tangent to the curve y=3/x.
Work it through to get 3x²-cx+3=0
Now, would the discriminant be = 0 because there are equal roots? Can somebody explain to me why the tangent has equal roots, is it because it's just a straight line, essentially?

Reply 1

TenOfThem

18

The statement that tangents have equal roots is nonsense

What you mean is

The equation tangent = curve has one root

A line will

(i) cross a line twice in which case you will have 2 x-values where they meet so line=curve would have 2 roots

(ii) touch the curve at one point, giving 1 x-value so here line=curve will have 2 repeated roots (or 1 root)

(iii) miss the curve completely do line=curve has no roots

Reply 2

TSR George

OP

17

Original post by TenOfThem

The statement that tangents have equal roots is nonsense

What you mean is

The equation tangent = curve has one root

A line will

(i) cross a line twice in which case you will have 2 x-values where they meet so line=curve would have 2 roots

(ii) touch the curve at one point, giving 1 x-value so here line=curve will have 2 repeated roots (or 1 root)

(iii) miss the curve completely do line=curve has no roots