Problem 39
Question
The differential equation of the family of circles with fixed radius 5 units and centre on the line \(y=2\) is (A) \((x-2) y^{\prime 2}=25-(y-2)^{2}\) (B) \((y-2) y^{\prime 2}=25-(y-2)^{2}\) (C) \((y-2)^{2} y^{2}=25-(y-2)^{2}\) (D) \((x-2)^{2} y^{\prime 2}=25-(y-2)^{2}\)
Step-by-Step Solution
Verified Answer
The correct option is (B).
1Step 1: Define the equation of the circle
Given that the radius of the circle is 5 units and the center is on the line \(y=2\), we can represent the equation of the family of circles as \((x-a)^2 + (y-2)^2 = 5^2\). Here, \(a\) is the x-coordinate of the center.
2Step 2: Differentiate the circle equation implicitly
Differentiate both sides of the equation \((x-a)^2 + (y-2)^2 = 25\) with respect to \(x\). This yields \(2(x-a) + 2(y-2)\frac{dy}{dx} = 0\). Simplifying gives \(\frac{dy}{dx} = -\frac{x-a}{y-2}\).
3Step 3: Eliminate the parameter 'a'
Use the fact that \((x-a)^2 + (y-2)^2 = 25\) to express \(x-a = \sqrt{25 - (y-2)^2}\). Substitute this into the differential equation \(\frac{dy}{dx} = -\frac{x-a}{y-2}\) to eliminate \(a\).
4Step 4: Substitute back to form the differential equation
Substituting \(x-a = \sqrt{25 - (y-2)^2}\) into \(\frac{dy}{dx} = -\frac{x-a}{y-2}\), we rearrange to get \((y-2)^2\left(\frac{dy}{dx}\right)^2 = 25 - (y-2)^2\).
5Step 5: Compare with given options
This expression matches option (B). Thus, the differential equation is \((y-2) y^{\prime 2}=25-(y-2)^{2}\).
Key Concepts
Family of CirclesImplicit DifferentiationRadius of a Circle
Family of Circles
In mathematics, a family of circles is a group of circles that share a common characteristic. For this exercise, the family of circles is defined by having a fixed radius of 5 units, and all the centers of these circles lie on the line \(y=2\). This line acts as a constraint from which all the centers originate.
One can imagine this as an infinite set of circles that "slide" along the line \(y=2\) while maintaining their size, as every circle in this family has the same radius of 5 units. The general equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. For our family of circles, the equation becomes \((x-a)^2 + (y-2)^2 = 5^2\), where \(a\) varies, reflecting the changing x-coordinate of the center.
Understanding such a family helps in solving problems where circles lie on a geometrical path or have constraints like lines or curves they must pass through.
One can imagine this as an infinite set of circles that "slide" along the line \(y=2\) while maintaining their size, as every circle in this family has the same radius of 5 units. The general equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. For our family of circles, the equation becomes \((x-a)^2 + (y-2)^2 = 5^2\), where \(a\) varies, reflecting the changing x-coordinate of the center.
Understanding such a family helps in solving problems where circles lie on a geometrical path or have constraints like lines or curves they must pass through.
Implicit Differentiation
Implicit differentiation is a technique in calculus used to find derivatives of equations not explicitly solved for one variable in terms of another. When dealing with circles, their equations are often not easily rearranged to solve directly, hence the need for implicit differentiation.
In our exercise, the circle's equation \((x-a)^2 + (y-2)^2 = 25\) is differentiated with respect to \(x\). Doing this directly involves treating \(y\) as a function of \(x\), or \(y(x)\). We apply the chain rule here, differentiating each term:
In our exercise, the circle's equation \((x-a)^2 + (y-2)^2 = 25\) is differentiated with respect to \(x\). Doing this directly involves treating \(y\) as a function of \(x\), or \(y(x)\). We apply the chain rule here, differentiating each term:
- Differentiating \((x-a)^2\) gives \(2(x-a)\);
- Differentiating \((y-2)^2\) gives \(2(y-2)\frac{dy}{dx}\);
- Setting the sum equal to zero reflects the equation's constant side, \(0\).
Radius of a Circle
The radius of a circle is the constant distance from its center to any point on the circle's edge. In this problem, all circles in the family share a fixed radius of 5 units. This constant radius is essential in formulating and solving the equation of the circle.
When constructing the equation for this family of circles, using the radius in the equation \((x-a)^2 + (y-2)^2 = 5^2\) is vital. Here, \(5\) is squared to emphasize that the equation form \(r^2\) is maintained.
The radius is a core aspect because it determines the size and boundary of the circles. In many mathematical problems, understanding how to work with the radius and its implications on the shape and properties of a circle is crucial. Knowing the radius allows us to predict and calculate other important circle properties, like area and circumference, although not directly required in this differential equation context.
When constructing the equation for this family of circles, using the radius in the equation \((x-a)^2 + (y-2)^2 = 5^2\) is vital. Here, \(5\) is squared to emphasize that the equation form \(r^2\) is maintained.
The radius is a core aspect because it determines the size and boundary of the circles. In many mathematical problems, understanding how to work with the radius and its implications on the shape and properties of a circle is crucial. Knowing the radius allows us to predict and calculate other important circle properties, like area and circumference, although not directly required in this differential equation context.
Other exercises in this chapter
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