To find the shortest distance between the parabola $y = (x - 5)^2 + 1$ and the circle $x^2 + y^2 = 4$, we can approach the problem using calculus and geometry. Here's a step-by-step outline:

Step 1: Understand the geometry

• The equation $y = (x - 5)^2 + 1$ represents a parabola shifted to the right by 5 units and 1 unit up.
• The equation $x^2 + y^2 = 4$ represents a circle centered at the origin with a radius of 2.

Our goal is to minimize the distance between any point $P_1 = (x_1, y_1)$ on the parabola and any point $P_2 = (x_2, y_2)$ on the circle.

Step 2: Distance formula

The distance between any two points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ is given by the formula: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

We aim to minimize this distance.

Step 3: Parametrize the points on the curve

• For the parabola $y_1 = (x_1 - 5)^2 + 1$.
• For the circle, we parametrize the points using polar coordinates. Let: $x_2 = 2 \cos \theta, \quad y_2 = 2 \sin \theta$ (where $\theta$ is a parameter that runs around the circle).

Step 4: Set up the distance function

The distance function $D$ between a point $P_1 = (x_1, y_1)$ on the parabola and a point $P_2 = (x_2, y_2)$ on the circle becomes: $D(x_1, \theta) = \sqrt{(2 \cos \theta - x_1)^2 + (2 \sin \theta - (x_1 - 5)^2 - 1)^2}$

Step 5: Minimize the distance function

To find the shortest distance, we can minimize the function $D(x_1, \theta)$ by taking partial derivatives with respect to $x_1$ and $\theta$, setting them equal to zero, and solving the resulting system of equations.

This process involves solving two simultaneous equations derived from the partial derivatives: $\frac{\partial D}{\partial x_1} = 0 \quad \text{and} \quad \frac{\partial D}{\partial \theta} = 0$

This step generally requires computational techniques due to the complexity of the expressions.

Would you like me to proceed with solving these equations using a computational approach?