**Solution:**

To find the ratio in which the plane divides the line joining the given points, we need to determine the point of intersection between the line and the plane. We can then calculate the ratio using the distances between the points.

**Finding the Point of Intersection:**

1. Let's first find the direction ratios of the line joining the two points. The direction ratios are given by:

*d₁ = x₂ - x₁ = 3 - (-2) = 5*

*d₂ = y₂ - y₁ = -5 - 4 = -9*

*d₃ = z₂ - z₁ = 8 - 7 = 1*

2. Now, let's find the equation of the line joining the two points using the direction ratios and a point on the line (say (-2, 4, 7)):

*x - x₁ / d₁ = y - y₁ / d₂ = z - z₁ / d₃*

*(x + 2) / 5 = (y - 4) / -9 = (z - 7) / 1*

Simplifying, we get:

*9x + 5y - 4z + 16 = 0* ...(1)

3. Now, let's find the equation of the given plane:

*x - 2y + 3z = 17* ...(2)

4. To find the point of intersection, we need to solve the equations (1) and (2) simultaneously. Let's solve them:

From equation (1), we can express x in terms of y and z:

*9x = -5y + 4z - 16*

*x = (-5y + 4z - 16) / 9*

Substituting this value of x in equation (2), we get:

*(-5y + 4z - 16) / 9 - 2y + 3z = 17*

Simplifying, we get:

*-5y + 4z - 16 - 18y + 27z = 153*

*-23y + 31z = 169* ...(3)

5. Now we have a system of two linear equations with two variables (equations (1) and (3)). Let's solve this system:

Multiplying equation (3) by 9, we get:

*-207y + 279z = 1521* ...(4)

Adding equations (1) and (4), we get:

*-198y + 275z = 1505* ...(5)

Solving equations (1) and (5), we find:

*y = -3*

Substituting this value of y in equation (1), we get:

*x = 2*