Question Statement

For what value(s) of $c$ does the line $2x+2y-c=0$ never touch or intersect the parabola $x^2-x+2y+3=0$?

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Background and Explanation

To determine the intersection of a line and a parabola, we solve their equations simultaneously to form a quadratic equation. If the line never touches or intersects the parabola, the resulting quadratic equation must have no real solutions, which occurs when its discriminant ($D=b^2-4ac$) is less than zero.

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Solution

Given the equation of the line: $2x+2y-c=0\dots \text{(i)}$

And the equation of the parabola: $x^2-x+2y+3=0\dots \text{(ii)}$

Step 1: Express $2y$ in terms of $x$ and $c$

From equation (i), we can isolate the term $2y$ to make substitution easier: $2y=c-2x$ $y=\frac{c-2x}{2}$

Step 2: Substitute into the parabola equation

Substitute the expression for $2y$ into equation (ii): $x^2-x+(c-2x)+3=0$

Now, simplify the equation by combining like terms: $x^2-x+c-2x+3=0$ $x^2-3x+(c+3)=0$

This is a quadratic equation in the form $a{x}^2+bx+k=0$, where:

• $a=1$
• $b=-3$
• $constant=c+3$

Step 3: Apply the condition for no intersection

The line will never touch or intersect the parabola if the discriminant of the quadratic equation is negative ($D<0$): $b^2-4ac<0$

Substitute the values of $a$, $b$, and the constant term: $(-3{)}^2-4(1)(c+3)<0$

Step 4: Solve the inequality for $c$

$9-4(c+3)<0$ $9-4c-12<0$ $-4c-3<0$ $-4c<3$

When dividing by a negative number, the inequality sign reverses: $c>\frac{3}{-4}$ $c>-\frac{3}{4}$

Thus, the line will never touch or intersect the parabola for all values of $c$ greater than $-\frac{3}{4}$.

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Key Formulas or Methods Used

• Substitution Method: Used to combine the linear and quadratic equations.
• Quadratic Form: $a{x}^2+bx+c=0$.
• Discriminant ($D$): $D=b^2-4ac$.
• Intersection Conditions:
  • $D>0$: Two points of intersection.
  • $D=0$: Tangent (touches at one point).
  • $D<0$: No intersection or contact.

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Summary of Steps

1. Rearrange the linear equation to solve for $2y$.
2. Substitute the expression for $2y$ into the parabola's equation.
3. Simplify the resulting equation into a standard quadratic form $a{x}^2+bx+c=0$.
4. Set the discriminant $b^2-4ac<0$ to satisfy the condition that the line never touches the parabola.
5. Solve the resulting inequality to find the range of values for $c$.