To solve the problem, let's first summarize the given information and then proceed with the steps to find the equation of the ellipse:

1. Center of the ellipse: $(0, 0)$ (origin).
2. Foci on the y-axis.
3. Eccentricity $e$: $\frac{\sqrt{3}}{2}$.
4. Distance from point $P(0, 3)$ to the ellipse: 4.

For an ellipse centered at the origin with its foci on the y-axis, the standard equation is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

where $b$ is the semi-major axis and $a$ is the semi-minor axis. The relationship between $a$, $b$, and the eccentricity $e$ is:

$e = \frac{c}{b}$

where $c$ is the distance from the center to a focus. Given $e = \frac{\sqrt{3}}{2}$:

$c = \frac{\sqrt{3}}{2} b$

Since the foci are on the y-axis, we have:

$c^2 = b^2 - a^2$

Combining the equations, we get:

$\left(\frac{\sqrt{3}}{2} b\right)^2 = b^2 - a^2$ $\frac{3}{4} b^2 = b^2 - a^2$ $a^2 = b^2 - \frac{3}{4} b^2$ $a^2 = \frac{1}{4} b^2$ $a = \frac{1}{2} b$

Next, we use the given information about the point $P(0, 3)$ and the maximum distance from this point to the ellipse being 4. The point lies on the y-axis, so it only affects the value of $b$:

The distance from $P(0, 3)$ to the ellipse along the y-axis is $|3| + b = 4$:

$3 + b = 4$ $b = 1$

So, $a$ is:

$a = \frac{1}{2} b = \frac{1}{2} \times 1 = \frac{1}{2}$

Therefore, the equation of the ellipse is:

$\frac{x^2}{\left(\frac{1}{2}\right)^2} + \frac{y^2}{1^2} = 1$ $\frac{x^2}{\frac{1}{4}} + y^2 = 1$ $4x^2 + y^2 = 1$

Thus, the equation of the ellipse is:

$4x^2 + y^2 = 1$

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you derive the standard form of the ellipse equation from its general form?
2. What is the relationship between the major and minor axes in an ellipse?
3. How do you determine the foci of an ellipse given its equation?
4. What is the geometric significance of the eccentricity of an ellipse?
5. Can you explain the method to find the distance from a point to an ellipse?

Tip: Always check if the ellipse is centered at the origin or not, as it simplifies the calculations significantly.