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

Question

UpStudy AI · Accepted Answer

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:**  
$$ k = 14 $$
$$ k = 14 $$

Determine the value of ’ ’ if the
line given by is a by
tangent to the function given by

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions

Determine the value of ’ ’ if the line given by is a by tangent to the function given by