Question

The order of differential equation of family of circles passing through intersection of \(L \equiv 3 x+4 y-7=0\) and \(S \equiv x^{2}+y^{2}-2 x-2 y+1=0\) is \((1) 1\) (2) 2 (3) 3 (4) 4

Short answer

The order of the differential equation for the family of circles passing through the intersection is 2.

Step-by-step solution

Find the intersection points

First, solve the linear equation, 3x + 4y - 7 = 0, and the circle equation, x^2 + y^2 - 2x - 2y + 1 = 0, simultaneously. This involves substituting one equation into the other to find the points where they intersect.

Derive the general circle equation

Use the intersection points to write the general form of the equation of the family of circles. Since the circles pass through these intersection points, their equation will be in the form of (x - a)^2 + (y - b)^2 = r^2.

Impose the condition of intersection

Substitute the points of intersection into the circle's equation to impose the conditions. This will result in a system of equations that can be used to eliminate the parameter.

Form the differential equation

To find the differential equation, differentiate the general form of the circle equation with respect to x enough times until it no longer contains arbitrary constants. Ensure to express everything in terms of x, y, and their derivatives.

Determine the order

Count the highest number of derivatives to determine the order of the differential equation. The highest order derivative present will indicate the order of the differential equation.

Key concepts

Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives. In the context of a family of curves, it's often useful to describe such curves using differential equations. The general form of a differential equation can involve derivatives of different orders. For example, if the function in question is denoted as y(x), then higher-order derivatives like \(y', y''\), etc., can appear in the equation.
When dealing with geometric problems like families of circles, differential equations provide a powerful tool to encapsulate the relationship among all circles that share certain properties. In this exercise, we're interested in finding the differential equation that describes all circles passing through the intersection of given curves.

Intersection of Curves

The intersection of curves refers to the points where two or more curves meet or cross each other. To find these points, we solve the equations of the curves simultaneously. In the given problem:
\[L \equiv 3x + 4y - 7 = 0\]
\[S \equiv x^2 + y^2 - 2x - 2y + 1 = 0\]
The first is a linear equation representing a straight line, and the second is a circle equation. To find their intersection, we solve these equations together. This typically involves substituting one equation into the other to form a single equation with one variable, which can then be solved to find the exact intersection points.

Order of Differential Equations

The 'order' of a differential equation is the highest derivative of the function that appears in the equation. For example, an equation involving \(y''\) (the second derivative of y) would be a second-order differential equation. The order provides insight into the complexity and behavior of the system described by the equation.
When forming a differential equation from a family of curves, the order is determined by how many times you need to differentiate the general equation of the curve (e.g., the circle) until all arbitrary constants are eliminated. In the given problem, after performing these differentiations, we count the highest order derivative present to determine the order of the resulting differential equation.

Family of Circles

A family of circles is a set of circles sharing a common property, such as passing through specific points or having a common center. In the given problem, the family of circles all pass through the intersection points of the given line and circle.
The general equation for a family of circles can be formed by considering a standard circle equation like \((x-a)^2 + (y-b)^2 = r^2\), and then modifying it to meet the given conditions (in this case, the intersection points). By imposing these conditions, we derive a system of equations that eventually yields the differential equation describing the family. This final differential equation succinctly describes all possible circles sharing the specified property.