Which line will have no solution with the parabola [tex]y - x + 2 = x^2[/tex]?

A. [tex]y = x + k[/tex], where [tex]k \neq 2[/tex]

B. [tex]y = -x + 2[/tex]

C. [tex]y = 2x - 2[/tex]

D. [tex]y = 0[/tex]



Answer :

To determine which line will have no solution with the parabola [tex]\( y - x + 2 = x^2 \)[/tex], we need to examine the points of intersection between the given parabola and a generic straight line given by [tex]\( y = mx + c \)[/tex].

We start by substituting [tex]\( y = mx + c \)[/tex] into the parabola equation:

[tex]\[ (mx + c) - x + 2 = x^2 \][/tex]

Simplifying, we obtain:

[tex]\[ mx + c - x + 2 = x^2 \][/tex]

which simplifies to:

[tex]\[ x^2 - (m+1)x + (c + 2) = 0 \][/tex]

This is a quadratic equation in [tex]\(x\)[/tex]:

[tex]\[ x^2 - (m+1)x + (c + 2) = 0 \][/tex]

For the line to have no intersection with the parabola, this quadratic equation must have no real solutions. For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the condition for having no real solutions is that the discriminant must be negative. The discriminant [tex]\(\Delta\)[/tex] of the quadratic equation [tex]\( x^2 - (m+1)x + (c + 2) = 0 \)[/tex] is:

[tex]\[ \Delta = b^2 - 4ac \][/tex]

Here, [tex]\(a = 1\)[/tex], [tex]\(b = -(m+1)\)[/tex], and [tex]\(c = c + 2\)[/tex]. Substituting these values into the discriminant formula, we get:

[tex]\[ \Delta = (-(m+1))^2 - 4 \cdot 1 \cdot (c + 2) \][/tex]

[tex]\[ \Delta = (m+1)^2 - 4(c + 2) \][/tex]

For there to be no real solutions:

[tex]\[ \Delta < 0 \][/tex]

So:

[tex]\[ (m+1)^2 - 4(c + 2) < 0 \][/tex]

[tex]\[ (m+1)^2 < 4(c + 2) \][/tex]

[tex]\[ (m+1)^2 < 4c + 8 \][/tex]

Solving for [tex]\(c\)[/tex]:

[tex]\[ (m+1)^2 < 4c + 8 \][/tex]

[tex]\[ 4c + 8 > (m+1)^2 \][/tex]

[tex]\[ 4c > (m+1)^2 - 8 \][/tex]

[tex]\[ c > \frac{(m+1)^2 - 8}{4} \][/tex]

Therefore, the line [tex]\( y = mx + c \)[/tex] will have no intersection with the parabola [tex]\( y - x + 2 = x^2 \)[/tex] if the constant [tex]\( c \)[/tex] satisfies:

[tex]\[ c > \frac{(m+1)^2 - 8}{4} \][/tex]

This inequality determines the values of [tex]\( c \)[/tex] for which the given line does not intersect with the parabola.