https://file.aiquickdraw.com/mathbot/file/1726969086597zwkfcan3.jpg

Let's solve the problem step by step.

### Step 1: Determine if the point \((-2, -3, 2)\) is visible from \((5, 6, 0)\)
- The opaque ball is centered at the origin \((0, 0, 0)\) with a radius of 1.
- To determine if \((-2, -3, 2)\) is visible from \((5, 6, 0)\), we need to check if the line segment connecting these two points intersects the sphere.

#### Equation of the line:
The line passing through \((5, 6, 0)\) and \((-2, -3, 2)\) can be parameterized as:
\[
\mathbf{r}(t) = (1-t) \mathbf{r_1} + t \mathbf{r_2}
\]
where \(\mathbf{r_1} = (5, 6, 0)\) and \(\mathbf{r_2} = (-2, -3, 2)\), and \(t\) is a parameter.

So the equation of the line is:
\[
\mathbf{r}(t) = (5 + t(-7), 6 + t(-9), 0 + t(2))
\]
which simplifies to:
\[
\mathbf{r}(t) = (5 - 7t, 6 - 9t, 2t)
\]

#### Step 2: Check for intersection with the sphere
The equation of the sphere is:
\[
x^2 + y^2 + z^2 = 1
\]

Substitute the parametric coordinates of the line into the sphere's equation:
\[
(5 - 7t)^2 + (6 - 9t)^2 + (2t)^2 = 1
\]
Expanding and solving this quadratic equation will tell us whether the line intersects the sphere.

### Step 3: Solve the quadratic equation
Expand and simplify:
\[
(5 - 7t)^2 = 25 - 70t + 49t^2
\]
\[
(6 - 9t)^2 = 36 - 108t + 81t^2
\]
\[
(2t)^2 = 4t^2
\]
Adding these up:
\[
(25 - 70t + 49t^2) + (36 - 108t + 81t^2) + 4t^2 = 1
\]
\[
25 + 36 - 1 - 70t - 108t + (49t^2 + 81t^2 + 4t^2) = 0
\]
\[
60 - 178t + 134t^2 = 0
\]

This is a quadratic equation in \(t\):
\[
134t^2 - 178t + 60 = 0
\]

Solve for \(t\) using the quadratic formula:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 134\), \(b = -178\), and \(c = 60\).

### Step 4: Determine the point on the sphere
If the discriminant \(b^2 - 4ac\) is positive, the line intersects the sphere, and we can calculate the intersection points. If it's negative, there's no intersection, and the point \((-2, -3, 2)\) is visible from \((5, 6, 0)\).

Let me calculate this quadratic equation and find the solution for you.The discriminant of the quadratic equation is \(-476\), which is negative. This means that the line segment connecting the points \((5, 6, 0)\) and \((-2, -3, 2)\) **does not** intersect the sphere. 

### Conclusion:
- The point \((-2, -3, 2)\) is **visible** from \((5, 6, 0)\).
- Since the point \((-2, -3, 2)\) is visible, the direction \((-2, -3, 2)\) does not intersect the sphere, and the intersection calculation isn't needed for finding another point.

Would you like more details or have any questions?

### Related Questions:
1. How do you find the intersection points between a line and a sphere?
2. What is the significance of the discriminant in quadratic equations?
3. How can you determine the visibility of a point relative to an object?
4. What are the steps to parameterize a line in 3D space?
5. How do you compute the direction vector between two points?

### Tip:
When dealing with visibility problems involving spherical objects, always start by checking the discriminant of the intersection equation to determine whether a line actually intersects the sphere.

Is the point (-2, -3, 2) visible from the point (5, 6, 0) if there is an opaque ball of radius 1 centered at the origin?

Explore the visibility problem involving an opaque sphere and points in 3D space. Learn how to calculate intersections using parametric equations and quadratic formulas to determine if a point is visible from another point. This solution involves the intersection of a line with a sphere and solving quadratic equations.

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