Let a, b `in` R and a `gt` 0. If the tangent at the point (2, 2) to the circle `x^2 + y^2 = 8` touches the parabola `y^2` = 4a(x- b), then b-a is equal to _____

Answer

To solve the problem, we need to find the value of \( b - a \) given the conditions of the circle and the parabola. Let's break down the solution step by step.

### Step 1: Find the equation of the tangent to the circle at the point (2, 2)

The equation of the circle is given by:
\[ x^2 + y^2 = 8 \]
The radius \( r \) is \( \sqrt{8} = 2\sqrt{2} \).

The formula for the equation of the tangent to a circle at a point \( (x_1, y_1) \) is:
\[ xx_1 + yy_1 = r^2 \]
Substituting \( (x_1, y_1) = (2, 2) \) and \( r^2 = 8 \):
\[ x \cdot 2 + y \cdot 2 = 8 \]
This simplifies to:
\[ 2x + 2y = 8 \quad \Rightarrow \quad x + y = 4 \]
Thus, the equation of the tangent line is:
\[ x + y = 4 \]

### Step 2: Write the equation of the parabola

The equation of the parabola is given by:
\[ y^2 = 4a(x - b) \]

### Step 3: Substitute \( x \) from the tangent equation into the parabola's equation

From the tangent equation \( x + y = 4 \), we can express \( x \) in terms of \( y \):
\[ x = 4 - y \]
Now substitute this into the parabola's equation:
\[ y^2 = 4a((4 - y) - b) \]
This simplifies to:
\[ y^2 = 4a(4 - b - y) \]
Rearranging gives:
\[ y^2 + 4ay = 16a - 4ab \]
or:
\[ y^2 + 4ay + (4ab - 16a) = 0 \]

### Step 4: Set the discriminant of the quadratic equation to zero

For the line to be tangent to the parabola, the discriminant of this quadratic equation must be zero:
\[ D = b^2 - 4ac = 0 \]
Here, \( a = 1 \), \( b = 4a \), and \( c = 4ab - 16a \). Thus, the discriminant is:
\[ (4a)^2 - 4 \cdot 1 \cdot (4ab - 16a) = 0 \]
This simplifies to:
\[ 16a^2 - 16ab + 64a = 0 \]
Dividing through by 16 gives:
\[ a^2 - ab + 4a = 0 \]

### Step 5: Factor the quadratic equation

Factoring gives:
\[ a(a - b + 4) = 0 \]
Since \( a > 0 \), we have:
\[ a - b + 4 = 0 \quad \Rightarrow \quad b = a + 4 \]

### Step 6: Find \( b - a \)

Now we can find \( b - a \):
\[ b - a = (a + 4) - a = 4 \]

### Final Answer

Thus, the value of \( b - a \) is:
\[ \boxed{4} \]