1. Core Principle: The Shortest Distance from a Point to a Curve

When seeking the minimum distance from a fixed point to a curve, a fundamental geometric principle is at play: the shortest distance occurs along the line segment that is normal to the curve at the point of closest approach. This means the line segment connecting the fixed point to the point on the curve must be perpendicular to the tangent of the curve at that point. Consequently, the line segment itself serves as the normal to the curve at that specific point.

In this problem, we need to find a point $P$ on the parabola $x^2 = 4y$ such that its distance from the center of the circle $x^2 + y^2 + 6x + 8 = 0$ is minimum. According to the principle, the line segment connecting $P$ to the center of the circle must be the normal to the parabola at $P$. This condition allows us to equate the slope of the line segment with the slope of the normal to the parabola, enabling us to find the coordinates of $P$.

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2. Step-by-step Solution

2.1. Understanding the Parabola

The given equation of the parabola is $x^2 = 4y$.

• Identification: This is a standard upward-opening parabola of the form $x^2 = 4ay$. By comparing, we can see that $4a = 4$, which implies $a=1$.
• General Point Parameterization: To represent any point $P$ on this parabola, we use a parametric form. For a parabola $x^2 = 4ay$, the general parametric coordinates are $(2at, at^2)$. Substituting $a=1$, the coordinates of point $P$ are $(2t, t^2)$.
• Why Parameterize? Using a single parameter $t$ (instead of $(x_0, y_0)$ with the constraint $x_0^2 = 4y_0$) simplifies calculations significantly. It allows us to express both $x$ and $y$ coordinates as functions of a single variable, making it much easier to find derivatives and solve for the specific point later. This method inherently ensures that the point $P$ always lies on the parabola.

2.2. Identifying the Center of the Circle

The given equation of the circle is $x^2 + y^2 + 6x + 8 = 0$.

• Standard Form: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$.
• Extraction of Center: By comparing the given equation with the general form:
  • The coefficient of $x$ is $2g = 6 \implies g = 3$.
  • There is no $y$ term, so the coefficient of $y$ is $2f = 0 \implies f = 0$.
  • The constant term is $c = 8$.
• Result: The center of the circle, let's denote it as $C$, is $(-g, -f) = (-3, 0)$.
• Tip: Always be careful with the signs when extracting the center coordinates from the general form of the circle equation. The radius of the circle is $r = \sqrt{g^2+f^2-c} = \sqrt{3^2+0^2-8} = \sqrt{9-8}=1$. While not directly needed for this problem (since we are minimizing distance to the center, not the circle itself), it's good practice to be aware of the full circle properties.

2.3. Applying the Minimum Distance Condition

We are seeking a point $P$ on the parabola such that its distance from the center of the circle