$5$ Find the values of the constant $c$ for which the line $4 y=2 x+c$ is a tangent to th curve $y=4 x+\frac{8}{x}$ .

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5

Find the values of the constant

c

for which the line

4 y=2 x+c

is a tangent to th curve

y=4 x+\frac{8}{x}

Set Equations Equal: To find the values of $c$ for which the line is tangent to the curve, we need to set the two equations equal to each other and find the points of intersection. A tangent line touches the curve at exactly one point, so we are looking for a single solution to the system of equations.
Write Line Equation: First, let's write the equation of the line in slope-intercept form, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. The given line is $4y = 2x + c$ , which can be rewritten as $y = \frac{1}{2}x + \frac{c}{4}$ .
Find Points of Intersection: Now, let's set the equation of the line equal to the equation of the curve to find the points of intersection. We have $y = 4x + \frac{8}{x}$ and $y = \frac{1}{2}x + \frac{c}{4}$ . Setting them equal gives us $4x + \frac{8}{x} = \frac{1}{2}x + \frac{c}{4}$ .
Clear Denominator: To solve for $x$ , we need to get rid of the fraction. Multiply every term by $x$ to clear the denominator: $x(4x + \frac{8}{x}) = x(\frac{1}{2}x + \frac{c}{4})$ . This simplifies to $4x^2 + 8 = \frac{1}{2}x^2 + \frac{c}{4}x$ .
Set Equation to Zero: Now, let's move all terms to one side to set the equation to zero: $4x^2 - \left(\frac{1}{2}\right)x^2 - \left(\frac{c}{4}\right)x - 8 = 0$ . Simplifying the $x^2$ terms gives us $\left(\frac{7}{2}\right)x^2 - \left(\frac{c}{4}\right)x - 8 = 0$ .
Discriminant Calculation: This is a quadratic equation in terms of $x$ . For the line to be tangent to the curve, this quadratic equation must have exactly one solution. This means the discriminant of the quadratic equation must be zero. The discriminant is given by $b^2 - 4ac$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.
Quadratic Equation: In our quadratic equation $(\frac{7}{2})x^2 - (\frac{c}{4})x - 8 = 0$ , $a = \frac{7}{2}$ , $b = -\frac{c}{4}$ , and $c = -8$ . Plugging these into the discriminant formula gives us $(-\frac{c}{4})^2 - 4\cdot(\frac{7}{2})\cdot(-8) = 0$ .
Discriminant Formula: Solving for $c$ , we have $\frac{c^2}{16} - 4\times\left(\frac{7}{2}\right)\times(-8) = 0$ . This simplifies to $\frac{c^2}{16} + 56 = 0$ . Multiplying through by $16$ to clear the fraction gives us $c^2 + 56\times16 = 0$ .
Solve for c: Now, we solve for $c^2$ : $c^2 = -56 \times 16$ . Since $c^2$ must be a non-negative number, there is no real solution for $c$ . This means there is a math error in our previous steps, as we should expect to find a real value for $c$ .