To solve this problem, we first need to determine the equation of the line that passes through the point $(2, 3, 0)$ in the direction of $(-4, -5, 2)$, and then find where this line intersects the sphere.

Step 1: Line Equation

The parametric equation of the line can be written as:

$(x, y, z) = (2, 3, 0) + t((-4 - 2), (-5 - 3), (2 - 0))$ $(x, y, z) = (2, 3, 0) + t(-6, -8, 2)$ So, the parametric equations of the line are:

$x = 2 - 6t$ $y = 3 - 8t$ $z = 0 + 2t = 2t$

Step 2: Equation of the Sphere

The equation of the sphere is given as:

$x^2 + y^2 + z^2 = 13$

Step 3: Substitution

Now, substitute the parametric equations of the line into the equation of the sphere:

$(2 - 6t)^2 + (3 - 8t)^2 + (2t)^2 = 13$

Step 4: Expand and Simplify

Let's expand and simplify the equation step by step.

1. $(2 - 6t)^2 = 4 - 24t + 36t^2$
2. $(3 - 8t)^2 = 9 - 48t + 64t^2$
3. $(2t)^2 = 4t^2$

Now combine them:

$4 - 24t + 36t^2 + 9 - 48t + 64t^2 + 4t^2 = 13$ $4 + 9 + 36t^2 + 64t^2 + 4t^2 - 24t - 48t = 13$ $13 + 104t^2 - 72t = 13$ Simplify further:

$104t^2 - 72t = 0$

Step 5: Solve the Quadratic Equation

Factor out $t$:

$t(104t - 72) = 0$

So, either $t = 0$ or $104t = 72$, which gives:

$t = \frac{72}{104} = \frac{18}{26} = \frac{9}{13}$

Step 6: Find the Coordinates

Now substitute $t = \frac{9}{13}$ into the parametric equations for $x$, $y$, and $z$:

$x = 2 - 6\left(\frac{9}{13}\right) = 2 - \frac{54}{13} = \frac{26}{13} - \frac{54}{13} = \frac{-28}{13} \approx -2.1538$ $y = 3 - 8\left(\frac{9}{13}\right) = 3 - \frac{72}{13} = \frac{39}{13} - \frac{72}{13} = \frac{-33}{13} \approx -2.5385$ $z = 2\left(\frac{9}{13}\right) = \frac{18}{13} \approx 1.3846$

Final Answer:

The point on the sphere is approximately:

$(x, y, z) = (-2.1538, -2.5385, 1.3846)$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find the parametric equation of a line in 3D space?
2. What happens if the point $(-4, -5, 3)$ is visible from $(2, 3, 0)$?
3. How can you derive the equation of a sphere from its geometric properties?
4. What are the steps to solving quadratic equations that result from intersections of lines and spheres?
5. How can parametric equations be used to model motion in 3D space?

Tip: Always simplify parametric equations before substituting into other expressions to minimize calculation errors!