Question

Find the range of values of k for which the line y = x - k meets the curve y = kx ^ 2 + 7x.​

Answer 1

Answer:

y=x-k

putting value of y

kx^2 + 7x = x-k

kx^2 +7x -x +k =0

kx^2 + 6x + k =0

kx^2 + k + 6x =0

k(x^2 + 1 ) + 6x =0

k(x^2 + 1 )

Answer 2

$\Huge\underline{\bold{Solution}}$

Consider the system equation,

$\hookrightarrow\begin{cases} & y=x-k \\ & y=kx^{2}+7x \end{cases}$

The points of intersection are found at the zeros of $kx^{2}+7x=x-k$.

$\hookrightarrow kx^{2}+6x+k=0$

For solutions to exist, the discriminant should be positive or zero.

$\hookrightarrow \dfrac{D}{4}=3^{2}-k^{2}\geq0$

Now, the two curves meet if and only if,

$\hookrightarrow k^{2}-3^{2}\leq0$

$\hookrightarrow \Large\red{\boxed{\red{\bold{-3\leq k\leq3}}}}$

$\Huge\underline{\bold{Answer}}$

Hence, the interval of $k$ such that $y=kx^{2}+7x$ and $y=x-k$ intersects is $-3\leq k\leq3$.