Question

find set of value of K for the curve whoch is y=kx^2-3x and the line y=x-k do not meet

Answer 1

$\large\underline{\sf{Solution-}}$

Given curve is

$\rm :\longmapsto\:y = {kx}^{2} - 3x - - - (1)$

and the equation of line is

$\rm :\longmapsto\:y = x - k - - - (2)$

Substituting the value of y from equation (2), to equation (1)

$\rm :\longmapsto\: {kx}^{2} - 3x = x - k$

$\rm :\longmapsto\: {kx}^{2} - 3x - x + k = 0$

$\rm :\longmapsto\: {kx}^{2} - 4x + k = 0$

Now, its a quadratic in x.

So, for no point of intersection, Discriminant < 0

$\rm :\implies\: {b}^{2} - 4ac < 0$

Here,

$\rm :\longmapsto\:a = k$

$\rm :\longmapsto\:b = - 4$

$\rm :\longmapsto\:c = k$

So, on substituting the values, we get

$\rm :\longmapsto\: {( - 4)}^{2} - 4(k)(k) < 0$

$\rm :\longmapsto\:16 - {4k}^{2} < 0$

$\rm :\longmapsto\:4 - {k}^{2} < 0$

$\rm :\longmapsto\:-({k}^{2} - 4) < 0$

$\rm :\longmapsto\:{k}^{2} - 4 > 0$

$\rm :\longmapsto\:(k - 2)(k + 2) > 0$

We know,

If a and b are positive real numbers such that a < b, and (x - a) (x - b) > 0, then x < a or x > b.

So, using this, we get

$\rm :\implies\:k < - 2 \: \: or \: \: k > 2$

$\bf\implies \:k \: \in \: ( - \: \infty , \: - \: 2) \: \cup \: (2, \: \infty )$

Verification :-

Let assume that k = 3

So, given curve and line can be rewritten as

$\rm :\longmapsto\:y = 3 {x}^{2} - 3x$

Point of intersection with x - axis.

On x axis, y = 0

So,

$\rm :\longmapsto\: {3x}^{2} - 3x = 0$

$\rm :\longmapsto\: {x}^{2} - x = 0$

$\rm :\longmapsto\:x(x - 3) = 0$

$\rm :\longmapsto\:x = 0 \: \: \: or \: \: \: x = 3$

Point of intersection with y- axis

On y - axis, x = 0

$\rm :\longmapsto\:y = 0$

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 0 \\ \\ \sf 3 & \sf 0 \end{array}} \\ \end{gathered}$

Consider, the equation of line

$\rm :\longmapsto\:y = x - 3$

On substituting x = 0, we get

$\rm :\longmapsto\:y = - 3$

and

On substituting y = 0, we get

$\rm :\longmapsto\:x = 3$

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf - 3 \\ \\ \sf 3 & \sf 0 \end{array}} \\ \end{gathered}$

See the attachment graph.

We concluded that, given curves don't intersect with each other.