The set of values of 'c' so that the equations `y=|x|+c andx^(2)+y^(2)-8|x|-9=0` have no solution is

Options

A. `(-oo,-3)uu(3,oo)`
B. (-3,3)
C. `(-oo,-5sqrt2)uu(5sqrt2,oo)`
D. `(-oo,-4-5sqrt(2))uu(5sqrt2-4,oo)`

Answer

To solve the problem, we need to find the set of values of \( c \) such that the equations \( y = |x| + c \) and \( x^2 + y^2 - 8|x| - 9 = 0 \) have no solutions.

### Step-by-Step Solution:

1. **Substitute \( y \) in the second equation**:
We start with the equations:
\[ y = |x| + c \]
\[ x^2 + y^2 - 8|x| - 9 = 0 \]
Substitute \( y \) into the second equation:
\[ x^2 + (|x| + c)^2 - 8|x| - 9 = 0 \]

2. **Expand the equation**:
Expanding \( (|x| + c)^2 \):
\[ (|x| + c)^2 = |x|^2 + 2c|x| + c^2 \]
Thus, the equation becomes:
\[ x^2 + |x|^2 + 2c|x| + c^2 - 8|x| - 9 = 0 \]
Since \( |x|^2 = x^2 \), we can combine like terms:
\[ 2x^2 + 2c|x| + c^2 - 8|x| - 9 = 0 \]

3. **Rearranging the equation**:
Rearranging gives:
\[ 2x^2 + (2c - 8)|x| + (c^2 - 9) = 0 \]

4. **Analyze for no solutions**:
For the quadratic equation in \( |x| \) to have no solutions, the discriminant must be less than zero. The standard form is:
\[ ax^2 + bx + c = 0 \]
Here, \( a = 2 \), \( b = 2c - 8 \), and \( c = c^2 - 9 \). The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting in our values:
\[ D = (2c - 8)^2 - 4(2)(c^2 - 9) \]

5. **Simplifying the discriminant**:
Expanding the discriminant:
\[ D = (4c^2 - 32c + 64) - (8c^2 - 72) \]
\[ D = 4c^2 - 32c + 64 - 8c^2 + 72 \]
\[ D = -4c^2 - 32c + 136 \]

6. **Setting the discriminant less than zero**:
We need:
\[ -4c^2 - 32c + 136 < 0 \]
Dividing the entire inequality by -4 (and reversing the inequality):
\[ c^2 + 8c - 34 > 0 \]

7. **Finding the roots**:
We find the roots of the quadratic equation \( c^2 + 8c - 34 = 0 \) using the quadratic formula:
\[ c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 8, c = -34 \):
\[ c = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-34)}}{2 \cdot 1} \]
\[ c = \frac{-8 \pm \sqrt{64 + 136}}{2} \]
\[ c = \frac{-8 \pm \sqrt{200}}{2} \]
\[ c = \frac{-8 \pm 10\sqrt{2}}{2} \]
\[ c = -4 \pm 5\sqrt{2} \]

8. **Finding the intervals**:
The roots are \( c_1 = -4 - 5\sqrt{2} \) and \( c_2 = -4 + 5\sqrt{2} \). The quadratic \( c^2 + 8c - 34 \) opens upwards, so it is positive outside the roots:
\[ c \in (-\infty, -4 - 5\sqrt{2}) \cup (-4 + 5\sqrt{2}, \infty) \]

### Final Answer:
The set of values of \( c \) for which the equations have no solution is:
\[ c \in (-\infty, -4 - 5\sqrt{2}) \cup (-4 + 5\sqrt{2}, \infty) \]