find-the-values-of-k-for-which-the-line-y-kx-3-does-not-meet-the-curve-y-2x-2-3x-k-

Question Number 113857 by ayenisamuel last updated on 15/Sep/20

f i n d t h e v a l u e s o f k f o r w h i c h t h e

l i n e y = k x - 3 d o e s n o t m e e t t h e c u r v e

y = 2 x^{2} - 3 x + k .

Answered by MJS_new last updated on 15/Sep/20

first, intersect them

k x - 3 = 2 x^{2} - 3 x + k

x^{2} - \frac{k + 3}{2} x + \frac{k + 3}{2} = 0

let x = t + \frac{k + 3}{4} and solve for t^{2}

t^{2} = \frac{(k - 5) (k + 3)}{16}

\Rightarrow if \frac{(k - 5) (k + 3)}{16} < 0 \Rightarrow no solution

\Rightarrow - 3 < k < 5 is the answer

Answered by 1549442205PVT last updated on 16/Sep/20

We see that

l i n e y = k x - 3 d o e s n o t m e e t t h e c u r v e

y = 2 x^{2} - 3 x + k if and only if the equation

{2 x}^{2} - 3 x + k = kx - 3 has no roots

\Leftrightarrow {2 x}^{2} - (3 + k) x + k + 3 = 0 has no roots

\Leftrightarrow Δ = {(3 + k)}^{2} - 4.2 . (k + 3) < 0

\Leftrightarrow k^{2} - 2 k - 15 < 0 \Leftrightarrow (k - 5) (k + 3) < 0

\Leftrightarrow - 3 < k < 5

Answer is : - 3 < k < 5