Question

The differential equation of the family of circles with fixed radius 5 units and centers on the line \(\mathrm{y}=2\) is: (A) \((\mathrm{y}-2)^{2}(\mathrm{dy} / \mathrm{dx})^{2}=25-(\mathrm{y}-2)^{2}\) (B) \((\mathrm{y}-2)(\mathrm{dy} / \mathrm{dx})^{2}=25-(\mathrm{y}-2)^{2}\) (C) \((x-2)(d y / d x)^{2}=25-(y-2)^{2}\) (D) \((\mathrm{x}-2)^{2}(\mathrm{dy} / \mathrm{dx})^{2}=25-(\mathrm{y}-2)^{2}\)

Short answer

(A) \[(y-2)^2(\frac{dy}{dx})^2 = 25 - (y-2)^2\]

Step-by-step solution

Write the equation for a general circle with its center on the line y=2

We know that the equation for a circle with its center at (a, b) and radius r is given by \[(x-a)^2 + (y-b)^2 = r^2\] Since the circles have a fixed radius 5 units and the centers are on the line y=2, we can write the equation of the circle as \[(x-a)^2 + (y-2)^2 = 25\]

Differentiate the equation

Now, we'll differentiate both sides of the equation with respect to x, using the implicit differentiation method. \[\frac{d}{dx}\left[(x-a)^2+(y-2)^2\right]=\frac{d}{dx}(25)\] Differentiating each term with respect to x, we get: \[2(x-a) + 2(y-2)\frac{dy}{dx} = 0\]

Solve the differential equation and find the correct answer option

Now, we'll eliminate the arbitrary constant a from the equation by solving for it. \(a = x - (y-2)\frac{dy}{dx}\) We can substitute this expression for a back into the original equation of the circle: \[(x - (x - (y-2)\frac{dy}{dx})) ^2 + (y-2)^2 = 25\] Now, by simplifying and isolating the \((\frac{dy}{dx})^2\) term, the differential equation can be written as: \[((y-2)^2)\left(\frac{dy}{dx}\right)^2 = 25 - (y-2)^2\] Comparing this differential equation with the given options, we find that it matches option (A). So, the correct answer is: (A) \[(y-2)^2(\frac{dy}{dx})^2 = 25 - (y-2)^2\]

Key concepts

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of an equation when it's not explicitly solved for one variable in terms of the others. This approach is particularly helpful when dealing with equations of curves, like circles, that aren't easily expressed as functions.

For example, the equation of a circle doesn't neatly resolve into a function of the form y = f(x), because for each x, there may be two corresponding y-values. In our case, the general equation for the circle is \[(x-a)^2 + (y-2)^2 = 25\]
, which implicitly defines y in terms of x. We differentiate term by term, treating y as a function of x (noted as y(x)), and apply the chain rule when differentiating terms involving y. The result is an equation that relates the derivative dy/dx to the variables x and y.

Family of Circles

A family of circles is a set of circles that share a common geometric property, such as having the same radius or centers that lie on a specific line. In this context, every circle in the family is represented by a general equation that contains one or more arbitrary constants. These constants can vary to produce different members of the family.

For our circled problem, the arbitrary constant 'a' in the equation \[(x-a)^2 + (y-2)^2 = 25\]
represents the x-coordinate of the center of each circle in the family, whereas 'b' is held constant at 2. Any value of 'a' yields a different circle, but all share the common properties of a fixed radius of 5 units and centers along the line y=2.

Fixed Radius Circles

Circles with fixed radius are a set where every circle has the same distance from the center point to any point on its circumference. This distance is the radius. For our example, the radius is given as 5 units, meaning that every circle in this family will have a radius of 5, regardless of where its center is located.

This is expressed in the circle's standard equation by setting the radius squared, r^2, equal to 25. The presence of this fixed radius in the equation ensures that the size of the circle, or its 'spread', remains constant, even as its position shifts based on the center's coordinates.

Centers on a Line

When we say that the centers of a family of circles lie on a specific line, what we mean is that the coordinate for the center of every circle in the family satisfies the equation of that line. In this scenario, the line specified is y = 2. This means that the y-coordinate for the center of each of our circles is fixed at 2.

From a geometry perspective, every possible circle with a center lying on this line can be thought of as sliding along the line, each keeping the same vertical position but varying in the horizontal direction. Mathematically, this is captured in the general circle equation where the y-coordinate of the center (b) is constantly 2, reflecting the condition that all the circle centers are indeed on the line y = 2.