题目

Problem

The straight line $l$ has equation $y=k(2x-1)$ , where $k$ is a constant.

The curve $C$ has equation $y=x^2+2x+11$ .

Find the set of values of $k$ for which $l$ does not cross or touch $C$ .

(6)

解答

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直线不与曲线相交或相切，表示联立后得到的二次方程没有实根。因此判别式要小于 $0$ 。

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At intersections,

\begin{align*} k(2x-1)=&\,x^2+2x+11\\ 2kx-k=&\,x^2+2x+11\\ x^2+(2-2k)x+(11+k)=&\,0. \end{align*}

For the line not to cross or touch the curve,

\begin{align*} (2-2k)^2-4(1)(11+k)&<0\\ 4-8k+4k^2-44-4k&<0\\ 4k^2-12k-40&<0\\ k^2-3k-10&<0\\ (k-5)(k+2)&<0. \end{align*}

Therefore

\begin{align*} -2<k<5. \end{align*}