The differential equation of the family of circles with fixed radius 5 units and centre on the line

Q: Why do we differentiate only once? Can't we differentiate twice?

Great question! The number of times we differentiate is strictly tied to the number of arbitrary constants. Since $h$ is the only parameter defining the family, one differentiation is sufficient to capture the 'rate of change' of the family and eliminate the constant.

Q: What happens to the $25$ when we differentiate?

Remember the fundamental rule of calculus: the derivative of a constant is always zero. Since the radius is fixed at 5, $r^2 = 25$ is a constant, so its derivative vanishes during the process.

Q: Is there a geometric interpretation of the final differential equation?

Yes! The differential equation represents the slope $y'$ at any point $(x, y)$ on any circle in the family. It tells us how the slope must behave to stay on a circle of radius 5 centered on $y = 2$.

Question

The differential equation of the family of circles with fixed radius 5 units and centre on the line $y = 2$ is

Options:

(A) $(x - 2) y'^2 = 25 - (y - 2)^2$

(B) $(y - 2) y'^2 = 25 - (y - 2)^2$

(C) $(y - 2)^2 y'^2 = 25 - (y - 2)^2$

(D) $(x - 2)^2 y'^2 = 25 - (y - 2)^2$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a coordinate plane, looking at a collection of circles. They are all the same size—a radius of $5$—but they are sliding horizontally, their centers locked onto the line $y = 2$.

In mathematics, a family is defined by a parameter that allows the curve to move or change shape. Here, that parameter is the $x$-coordinate of the center, which we call $h$.

The equation of any circle in this family is:
$$(x - h)^2 + (y - 2)^2 = 25$$

Our goal is to find the differential equation that governs this entire family. We want an equation that describes the slope $y'$ at any point $(x, y)$ on any of these circles, without needing to know the specific value of $h$.

### The Calculus of Elimination

To find the differential equation, we must exorcise the 'ghost' parameter $h$. Since there is only one arbitrary constant, we differentiate the equation exactly once with respect to $x$.

Applying the power rule and the chain rule:
$$\frac{d}{dx}[(x - h)^2] + \frac{d}{dx}[(y - 2)^2] = \frac{d}{dx}[25]$$

This yields:
$$2(x - h) + 2(y - 2)y' = 0$$

Simplifying this by dividing by $2$, we get:
$$(x - h) + (y - 2)y' = 0$$

This is a beautiful, simple relationship. It tells us that the horizontal distance from the center $(x - h)$ is directly related to the slope $y'$ and the vertical distance from the line $y = 2$.

### The Final Substitution

Now, we perform the final algebraic dance. We isolate $(x - h)$ to get:
$$(x - h) = -(y - 2)y'$$

We then take this expression and plug it back into our original circle equation. Substituting this into $(x - h)^2 + (y - 2)^2 = 25$, we get:
$$[-(y - 2)y']^2 + (y - 2)^2 = 25$$

Expanding the squared term, the negative sign vanishes, leaving us with:
$$(y - 2)^2 (y')^2 + (y - 2)^2 = 25$$

Rearranging to match the standard form, we arrive at the final differential equation:
$$(y - 2)^2 (y')^2 = 25 - (y - 2)^2$$

This is the differential equation of our family of circles. It is elegant, precise, and captures the essence of the entire family in one stroke. You have successfully navigated the transition from geometry to calculus!

Animated Solution for Mathematics - Differential Equations: The differential equation of the family of circles with fixed radius 5 units and centre on the line $y = 2$ is

Select Answer:

Visualized Solution

Visualizing the Family of Circles

Defining the Center Coordinates

Standard Equation of a Circle

Substituting Center and Radius

The Goal: Eliminate the Parameter $h$

Differentiating with Respect to $x$

Simplifying the Derivative

Isolating the Term $(x - h)$

Substituting Equation 2 into Equation 1

Expanding the Squared Term

Rearranging the Final Equation

Identifying the Correct Option

The Sigma Insight: Formation of Differential Equations

Analyzing the Setup

The Calculus of Elimination

The Final Substitution

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Animated Solution for Mathematics - Differential Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The differential equation of the family of circles with fixed radius 5 units and centre on the line y=2 is