Suppose that you stand at the point (2,3,0) and look in the direction of a point that is not visible because it is behind the ball. You will then be looking at a point on the sphere.

If (−4,−5,3) is not visible from (2,3,0), find the point on the sphere at which you are looking if you look in the direction of (−4,−5,3). Otherwise, find the point on the sphere at which you look if you are looking in the direction of (−4,−5,2).
Point (x,y,z)? The equation of sphere is x²+y²+z²=sqrt(13) ,please give answer round off to 5 d.p.

We are tasked with finding the point on the surface of a sphere that you are looking at, given that you are standing at the point \( P_0 = (2, 3, 0) \) and looking in the direction of another point.

### Key Information:
- The sphere has an equation \( x^2 + y^2 + z^2 = \sqrt{13} \).
- The given point is either \( (-4, -5, 3) \) (if not visible, the direction changes to \( (-4, -5, 2) \)).

Let’s break it down.

### Equation of Sphere
The equation given for the sphere seems incorrect as it is not a typical sphere equation. Assuming it was meant to be \( x^2 + y^2 + z^2 = 13 \) (standard form of a sphere, where \( 13 \) is the square of the radius, which is \( \sqrt{13} \)).

### Parametric Line Equation
The direction of the line you are looking in, from \( P_0 \), can be described parametrically as:

\[
\mathbf{r}(t) = (2, 3, 0) + t((-4 - 2), (-5 - 3), (3 - 0))
\]

This simplifies to:

\[
\mathbf{r}(t) = (2, 3, 0) + t(-6, -8, 3)
\]

Thus, the parametric equation for the line is:

\[
\mathbf{r}(t) = (2 - 6t, 3 - 8t, 3t)
\]

### Intersection with the Sphere
Substitute \( x = 2 - 6t \), \( y = 3 - 8t \), and \( z = 3t \) into the sphere equation \( x^2 + y^2 + z^2 = 13 \):

\[
(2 - 6t)^2 + (3 - 8t)^2 + (3t)^2 = 13
\]

Now, expand and simplify:

\[
(4 - 24t + 36t^2) + (9 - 48t + 64t^2) + 9t^2 = 13
\]
\[
4 + 9 + 36t^2 + 64t^2 + 9t^2 - 24t - 48t = 13
\]
\[
13 + 109t^2 - 72t = 13
\]
\[
109t^2 - 72t = 0
\]
\[
t(109t - 72) = 0
\]

So, \( t = 0 \) or \( t = \frac{72}{109} \).

- \( t = 0 \) gives the point \( (2, 3, 0) \), which is the starting point.
- \( t = \frac{72}{109} \) gives the point on the sphere.

### Coordinates of the Intersection Point
Substitute \( t = \frac{72}{109} \) into the parametric line equations:

\[
x = 2 - 6 \times \frac{72}{109} = 2 - \frac{432}{109} = \frac{218 - 432}{109} = \frac{-214}{109} \approx -1.96330
\]
\[
y = 3 - 8 \times \frac{72}{109} = 3 - \frac{576}{109} = \frac{327 - 576}{109} = \frac{-249}{109} \approx -2.28440
\]
\[
z = 3 \times \frac{72}{109} = \frac{216}{109} \approx 1.98165
\]

### Final Answer
The point on the sphere where you are looking is approximately \( (-1.96330, -2.28440, 1.98165) \).

---

Would you like further details or have any questions? Here are 5 follow-up questions related to this problem:

1. How is the parametric equation of a line derived in general?
2. What happens when you plug different values of \( t \) into the parametric equation?
3. What are some applications of finding intersection points with spheres?
4. How would the process change if the sphere had a different radius?
5. Can this method be used for other conic sections, like ellipses?

**Tip:** Always verify the form of a given equation before starting, as errors can propagate through calculations.

Suppose that you stand at the point (2,3,0) and look in the direction of a point that is not visible because it is behind the ball. You will then be looking at a point on the sphere. If (−4,−5,3) is not visible from (2,3,0), find the point on the sphere at which you are looking if you look in the direction of (−4,−5,3). Otherwise, find the point on the sphere at which you look if you are looking in the direction of (−4,−5,2). Point (x,y,z)? The equation of sphere is x²+y²+z²=sqrt(13), please give answer round off to 5 d.p.

Discover how to find the intersection point of a line and a sphere using parametric equations and quadratic formulas. This detailed solution addresses a geometry problem where a person looks in the direction of a point behind a sphere. Learn key concepts and equations used in undergraduate-level mathematics.

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