Determine the value of ' \( k \) ' if the line given by \( y=-2 x+k \) is a by tangent to the function given by \( y=-x^{2}+4 x+5 \)

To determine the value of \( k \) such that the line \( y = -2x + k \) is tangent to the parabola \( y = -x^2 + 4x + 5 \), follow these steps: 1. **Set the Equations Equal:** For the line to be tangent to the parabola, they must intersect at exactly one point. Set the two equations equal to each other: \[ -2x + k = -x^2 + 4x + 5 \] 2. **Rearrange the Equation:** Bring all terms to one side to form a quadratic equation: \[ x^2 - 6x + (k - 5) = 0 \] 3. **Condition for Tangency:** A quadratic equation \( ax^2 + bx + c = 0 \) has exactly one real solution (i.e., the line is tangent to the parabola) if its discriminant \( D \) is zero. The discriminant is given by: \[ D = b^2 - 4ac \] For the quadratic \( x^2 - 6x + (k - 5) = 0 \): \[ a = 1,\quad b = -6,\quad c = k - 5 \] So, \[ D = (-6)^2 - 4(1)(k - 5) = 36 - 4(k - 5) \] 4. **Solve for \( k \):** Set the discriminant to zero: \[ 36 - 4(k - 5) = 0 \] \[ 36 - 4k + 20 = 0 \] \[ 56 - 4k = 0 \] \[ 4k = 56 \] \[ k = 14 \] **Answer:** \( k = 14 \)