There is a point (p,q) on the graph of `f(x)=x^2` and a point `(r , s)` on the graph of `g(x)=(-8)/x ,w h e r ep >0a n dr > 0.` If the line through `(p , q)a n d(r , s)` is also tangent to both the curves at these points, respectively, then the value of `P+r` is_________.

Answer

To solve the problem, we need to find the value of \( p + r \) given the conditions about the points on the curves \( f(x) = x^2 \) and \( g(x) = -\frac{8}{x} \).

### Step 1: Identify the points on the curves
The point \( (p, q) \) lies on the graph of \( f(x) = x^2 \). Therefore, we can express \( q \) in terms of \( p \):
\[ q = f(p) = p^2 \]

The point \( (r, s) \) lies on the graph of \( g(x) = -\frac{8}{x} \). Thus, we can express \( s \) in terms of \( r \):
\[ s = g(r) = -\frac{8}{r} \]

### Step 2: Find the slopes of the tangents
Next, we find the derivatives of both functions to determine the slopes of the tangents at points \( (p, q) \) and \( (r, s) \).

The derivative of \( f(x) \) is:
\[ f'(x) = 2x \]
At \( x = p \):
\[ f'(p) = 2p \]

The derivative of \( g(x) \) is:
\[ g'(x) = \frac{8}{x^2} \]
At \( x = r \):
\[ g'(r) = \frac{8}{r^2} \]

### Step 3: Set the slopes equal
Since the line through the points \( (p, q) \) and \( (r, s) \) is tangent to both curves, the slopes must be equal:
\[ 2p = \frac{8}{r^2} \]

### Step 4: Solve for \( p \) in terms of \( r \)
Rearranging the equation gives:
\[ p = \frac{4}{r^2} \]

### Step 5: Find the slope of the line through the two points
The slope of the line through the points \( (p, q) \) and \( (r, s) \) can be expressed as:
\[ \text{slope} = \frac{s - q}{r - p} \]
Substituting for \( s \) and \( q \):
\[ \text{slope} = \frac{-\frac{8}{r} - p^2}{r - p} \]

### Step 6: Set the slopes equal
Setting this equal to \( 2p \):
\[ \frac{-\frac{8}{r} - p^2}{r - p} = 2p \]

### Step 7: Substitute \( p \) into the equation
Substituting \( p = \frac{4}{r^2} \) into the equation:
\[ \frac{-\frac{8}{r} - \left(\frac{4}{r^2}\right)^2}{r - \frac{4}{r^2}} = 2\left(\frac{4}{r^2}\right) \]

### Step 8: Simplify and solve for \( r \)
After simplifying, we can find the value of \( r \).

1. The left-hand side simplifies to:
\[ -\frac{8}{r} - \frac{16}{r^4} \]
and the denominator becomes:
\[ r - \frac{4}{r^2} = \frac{r^3 - 4}{r^2} \]

2. Setting the equation and solving gives us \( r = 1 \).

### Step 9: Find \( p \)
Substituting \( r = 1 \) back into \( p = \frac{4}{r^2} \):
\[ p = \frac{4}{1^2} = 4 \]

### Step 10: Calculate \( p + r \)
Finally, we calculate:
\[ p + r = 4 + 1 = 5 \]

### Final Answer
The value of \( p + r \) is \( \boxed{5} \).